Free Math Worksheets

For each binomial probability expression, identify the values of \(n\), \(p\), and \(k\). a) \(P(X=11)=\binom{20}{11}(0.75)^{11}(0.25)^9\) b) \(P(X=2)=\binom{50}{2}(0.05)^2(0.95)^{48}\) c) \(P(X=5)=\binom{9}{5}\left(\frac14\right)^5\left(\frac34\right)^4\)

A binomial experiment has \(n=16\), \(p=\frac13\), and \(k=4\). a) Write the complete expression for \(P(X=4)\). b) State the exponent on the failure probability and explain how it is found from \(n\) and \(k\).

A binomial probability is written as \(P(X=6)=\binom{15}{k}p^6(0.75)^m\). Find the missing values \(k\), \(p\), and \(m\), and then evaluate the probability to four decimal places.

Complete each binomial probability expression by filling in the missing values. a) \(P(X=3)=\binom{10}{\square}(0.4)^{\triangle}(0.6)^{\bigcirc}\) b) \(P(X=\square)=\binom{n}{5}\left(\frac13\right)^5(\triangle)^7\) c) \(P(X=2)=\binom{\square}{2}p^2(1-p)^8\)

Let \(X\) be binomial with \(n=12\) and \(p=\frac14\). Find \(P(X=4)\). Round to four decimal places.

An archer hits the target independently with probability \(0.80\) on each shot. The archer takes \(12\) shots. Find the probability of each event. a) Exactly \(10\) hits b) Exactly \(8\) hits c) No hits

Two factory machines produce components. Machine A produces a nondefective component with probability \(0.90\), while Machine B does so with probability \(0.70\). Assume component outcomes are independent for each machine. A sample of \(5\) components is taken from each machine. For which machine is it more likely that exactly \(4\) of the \(5\) components are nondefective? Support your answer with calculations.

Two archers, Jordan and Leah, each take \(3\) independent shots. Jordan hits the center with probability \(0.50\), while Leah hits the center with probability \(0.75\). Which event is more likely: Jordan gets exactly \(2\) center hits, or Leah gets center hits on all \(3\) shots?

A basketball player makes each free throw independently with probability \(p=0.6\). During practice, the player takes \(5\) free throws. Use the binomial probability formula to find the probability of making exactly \(3\) shots.

In a manufacturing process, \(20\%\) of components are defective. A sample of \(10\) components is selected, and the selections can be modeled as independent Bernoulli trials. Find the probability that exactly \(2\) components are defective.

Each LED light bulb produced by a manufacturer independently has a \(5\%\) chance of having a slight color variation. A quality-control inspector selects \(30\) bulbs. a) Find the probability that none of the bulbs in the sample has a color variation. b) Find the probability that at most one bulb in the sample has a color variation.

A biathlete hits the target on a standing shot with probability \(0.85\). Assume the outcomes of the shots are independent. During one shooting stage, she takes \(5\) shots. a) What is the probability that she makes exactly \(4\) hits? b) What is the probability that she makes at least \(4\) hits?

An archer independently takes \(4\) shots and hits the center of a target with probability \(0.60\) on each shot. Find the probability of exactly \(2\) center hits.

A gardener knows that each seed of a certain flower variety germinates independently with probability \(0.80\). The gardener plants \(7\) seeds. Find the probability that exactly \(5\) seeds germinate.

Use \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\) to calculate each binomial probability. Round to four decimal places when needed. a) \(P(X=3)\) for \(n=5\) and \(p=0.4\) b) \(P(X=2)\) for \(n=6\) and \(p=0.1\) c) \(P(X=0)\) for \(n=4\) and \(p=0.5\)

A binomial random variable has \(n=10\) and \(p=0.2\). Determine whether \(X=1\) or \(X=2\) is more likely.

The probability of an event \(A\) is given by \(P(A)=\binom{12}{2}\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^{10}\). Describe a random experiment that this formula could model, and state event \(A\) in words.

Use a calculator to evaluate each binomial probability. Round each answer to four decimal places. a) \(P(X=3)\) when \(n=15\) and \(p=0.2\) b) \(P(X=10)\) when \(n=20\) and \(p=0.45\) c) \(P(X=2)\) when \(n=50\) and \(p=0.05\)

A fair six-sided die is rolled \(10\) times. Use a calculator to find each probability. Round to four decimal places. a) Exactly two rolls show a \(6\). b) Exactly five rolls show a \(6\). c) No rolls show a \(6\).

A fair four-sided die, numbered \(1\) through \(4\), is rolled \(8\) times. Let \(X\) be the number of times a \(3\) is rolled. a) Find the probability \(p\) of a success, rolling a \(3\), on one roll. b) Use the binomial probability formula to find \(P(X=2)\). Show the binomial coefficient and the two powers before giving the final probability.

A city survey reports that \(65\%\) of residents regularly ride a bicycle. A random sample of \(100\) residents is selected, and the count can be modeled with a binomial random variable \(X\). Use a calculator to find the probability that exactly \(60\) sampled residents regularly ride a bicycle. Round to four decimal places.

At a high school, \(80\%\) of students usually eat breakfast before school. A random sample of \(6\) students is selected. The table shows probabilities for the number of sampled students who ate breakfast before school. Use the table to find the probability of each event: - Event A: Exactly \(5\) or \(6\) students ate breakfast. - Event B: At least \(4\) students ate breakfast. - Event C: At most \(3\) students ate breakfast. <table> <thead> <tr> <th>Number of students \(k\)</th> <th>Probability \(P(X=k)\)</th> </tr> </thead> <tbody> <tr> <td>\(6\)</td> <td>\(0.2621\)</td> </tr> <tr> <td>\(5\)</td> <td>\(0.3932\)</td> </tr> <tr> <td>\(4\)</td> <td>\(0.2458\)</td> </tr> </tbody> </table>

A random variable \(X\) has a binomial distribution with \(n=3\) and \(p=0.4\). Create a complete probability distribution table for \(X\).

A random variable \(X\) has a binomial distribution with \(n=5\) and \(p=0.2\). a) Calculate \(P(X=k)\) for every \(k\in\{0,1,2,3,4,5\}\). b) Check the distribution by adding all the individual probabilities.

An urn contains \(5\) red balls and \(3\) yellow balls. Two balls are drawn with replacement. The random variable \(X\) is the number of red balls drawn. a) Create the probability distribution of \(X\). b) Find the probability that at least one red ball is drawn.

A spinner has a red section that covers exactly one-fourth of the spinner. The spinner is spun independently \(6\) times. a) Find the probability that only the first two spins land on red. b) Find the probability that exactly two of the \(6\) spins land on red. c) Find the probability that none of the spins lands on red.

A multiple-choice test has \(15\) questions. Each question has four answer choices, exactly one of which is correct. A student guesses independently and randomly on every question. Find the probability that the student answers exactly \(5\) questions correctly.

A quiz has \(10\) multiple-choice questions. Each question has five answer choices, exactly one of which is correct. A student guesses independently and randomly on every question. a) How many correct-or-incorrect answer patterns contain exactly three correct answers? b) Find the probability that the student answers exactly three questions correctly. c) Find the probability that only the first three answers are correct and all remaining answers are incorrect.

A spinner has three equal sections colored red, blue, and green. It is spun independently \(8\) times. a) Find the probability that exactly \(3\) spins land on red. b) Find the probability that exactly \(6\) spins land on either red or blue.

In a manufacturing process, the probability that a microchip is defective is \(5\%\), and chip statuses are independent. A nondefective chip is labeled \(N\), and a defective chip is labeled \(D\). Eight chips are inspected in order. Consider these three inspection records: (1) \((N, N, N, N, N, N, N, N)\) (2) \((D, N, N, N, N, N, N, N)\) (3) \((D, D, D, D, D, D, D, D)\) a) Which record has the smallest probability? Explain without calculating. b) Calculate the probabilities of records (1) and (2). c) Someone claims, “The probability that exactly one chip is defective equals the probability of record (2).” Evaluate this claim mathematically.

An archer hits a target with probability \(p=0.8\) on each shot. The archer takes \(4\) independent shots. a) Find the probability of exactly two hits. b) Find the probability that only the first and fourth shots are hits.

Determine whether each situation can be modeled as a binomial experiment. Justify your decision. For each binomial experiment, give the number of trials \(n\) and the success probability \(p\). a) A container holds \(40\) screws, \(4\) of which are defective. Five screws are selected without replacement and checked for defects. b) A factory produces electronic components with a defect probability of \(0.5\%\). One hundred components are selected from ongoing production and tested, and their defect statuses are assumed independent. c) A fair six-sided die is rolled \(8\) times. A success is an even result.

Decide whether each situation is appropriately modeled as a binomial experiment. For each binomial experiment, give \(n\) and \(p\). a) A multiple-choice test has \(10\) questions, each with \(4\) answer choices and exactly one correct answer. A student guesses independently on every question. b) A committee of \(3\) students is selected without replacement from a class with \(12\) girls and \(13\) boys. A success is selecting a girl. c) A spinner has three equal sectors colored red, yellow, and blue. It is spun \(15\) times. A success is landing on yellow.

A gardener plants \(15\) seeds of a particular flower variety. Each seed germinates independently with probability \(85\%\). Find the probability of each event. a) Exactly \(13\) of the \(15\) seeds germinate. b) The first \(13\) seeds germinate, and the last two do not.

A basketball player makes a free throw with probability \(0.75\). The player takes \(12\) free throws. Briefly state the assumptions needed to model the series as a binomial setting, and find the probability that the player makes exactly \(9\) free throws.

A multiple-choice test has \(6\) questions. Each question has three answer choices, exactly one of which is correct. A student guesses independently and randomly on every question. Explain why \(\left(\frac13\right)^2\left(\frac23\right)^4\) is not the probability that the student answers exactly two questions correctly.

In a manufacturing process, \(90\%\) of the components are nondefective. A sample of \(6\) components is selected, and the outcomes are modeled as independent Bernoulli trials. 1. Find the probability that only the last selected component is defective. 2. Find the probability that exactly one component in the sample is defective. 3. Find the probability that the first three selected components are nondefective.

A fair six-sided die is rolled \(15\) times, and the rolls are independent. Let \(X\) be the number of \(6\)s rolled. Find each probability to four decimal places. 1. \(P(X=3)\) 2. \(P(X\le 4)\) 3. \(P(2\le X\le 5)\)

For a binomial random variable \(X\), define the cumulative distribution function by \(F(k)=P(X\le k)\). Use this definition to explain why \(F(k)-F(k-1)=P(X=k)\) for \(k\ge 1\).

For each expression, describe the probability it represents and calculate its value to four decimal places. a) \(\sum_{i=0}^{2}\binom{15}{i}(0.1)^i(0.9)^{15-i}\) b) \(1-\sum_{i=0}^{1}\binom{5}{i}(0.4)^i(0.6)^{5-i}\)

In a manufacturing process, a microchip is defective with probability \(p=0.05\). A sample of \(8\) chips is selected, and chip statuses are independent. For each expression, describe an event in context whose probability it represents. a) \((0.05)^2(0.95)^6\) b) \(\binom82(0.05)^2(0.95)^6\) c) \(1-(0.95)^8\)

A basketball player independently takes \(4\) free throws and makes each free throw with probability \(0.60\). Write an expression for the probability of each event. a) Only the first two shots are made. b) Exactly two shots are made. c) More than two shots are made.

An electronics manufacturer reports that \(4\%\) of the parts in a large shipment are defective. A customer randomly selects \(15\) parts. Assume the selections can be modeled as independent Bernoulli trials. Let \(X\) be the number of defective parts in the sample. Find the probability that the sample contains at least \(2\) defective parts.

A manufacturer ships microchips in packages of \(20\). Each chip independently has a \(10\%\) probability of being defective. A package is rejected during quality control if it contains more than \(3\) defective chips. What percentage of packages should be expected to be rejected?

A garden center sells packets containing \(12\) seeds of a certain flower variety. Each seed independently has a \(75\%\) probability of germinating. A customer considers a packet defective if fewer than \(8\) seeds germinate. Find the probability that a randomly selected packet is considered defective.

A biathlon athlete independently takes \(5\) shots and hits a target with probability \(p=0.8\) on each shot. Let \(X\) be the number of hits. A second athlete independently takes \(5\) shots and hits with probability \(0.2\) on each shot. Let \(Y\) be the number of hits. a) Calculate \(P(X=4)\) and \(P(Y=1)\). b) Without additional numerical calculations, explain why \(P(X=k)=P(Y=5-k)\) for every \(k\in\{0, 1, \ldots, 5\}\).

Determine whether the random variable \(X\) in each situation can be modeled with a binomial distribution. Justify your decision and state any needed assumptions. a) A bowl contains \(5\) red candies and \(15\) blue candies. A child takes \(4\) candies at once. The random variable \(X\) is the number of red candies selected. b) A spinner with three equal sectors—red, green, and blue—is spun \(10\) times. The random variable \(X\) is the number of spins that land on red. c) In a city with \(500{,}000\) households, about \(12\%\) own an electric vehicle. A random sample of \(50\) households is surveyed. The random variable \(X\) is the number in the sample that own an electric vehicle.

In a city survey, each person independently has a \(60\%\) chance of reporting that they regularly commute to work by bicycle. A group of \(15\) people is selected. Find the probability that a majority of the group regularly commutes by bicycle.

At a distribution center, \(5\%\) of packages are routed to the wrong facility. At the end of a day, an inspector randomly selects \(40\) packages. a) State the conditions under which the number of misrouted packages in the sample can be modeled with a binomial distribution. b) Define a suitable random variable \(X\), identify the parameters \(n\) and \(p\), and explain them in context. c) Find the probability that exactly two sampled packages were misrouted.

In a large forest, \(80\%\) of the trees are pine trees. A forester inspects a sample of \(200\) trees. Assume the sample can be modeled as independent Bernoulli trials, and let \(X\) be the number of pine trees in the sample. a) Find the probability of each event. \(E\): Exactly \(165\) of the inspected trees are pine trees. \(F\): From \(155\) through \(170\) of the inspected trees, inclusive, are pine trees. b) Describe the meaning of \(1-P(X\ge 175)\) in context.

A multiple-choice test has \(15\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A test taker guesses independently and randomly on every question. Write an expression for the probability of each event. a) Exactly \(6\) questions are answered correctly. b) At most \(1\) question is answered correctly. c) At least \(3\) questions are answered correctly.

A manufacturer finds that each LED diode independently has a \(1\) in \(20\) chance of being defective. a) How many defective diodes should the manufacturer expect in a daily production run of \(1400\) diodes? b) Find the probability that exactly \(1\) diode is defective in a random sample of \(25\) diodes. c) Find the probability that at most \(2\) diodes are defective in a random sample of \(25\) diodes.

A basketball player makes free throws with probability \(0.75\). The player takes \(45\) free throws. a) State the conditions needed to model the number of made free throws with a binomial distribution. b) Find the most likely number of made free throws.

A random variable \(X\) has a binomial distribution with \(n=80\) and \(p=0.45\). Without calculating the individual probabilities, justify the value of \(k\) for which \(P(X=k)\) is greatest.

Determine whether each experiment can be modeled as binomial. Give \(n\) and \(p\) when appropriate. a) An archer takes \(12\) shots. The probability of hitting the bull’s-eye is constant at \(0.80\), and the shots are treated as independent. Each shot is recorded as a hit or miss. b) Five cards are drawn without replacement from a well-shuffled standard \(52\)-card deck. Each draw is recorded as an ace or not an ace.

In a manufacturing process, each component independently has a \(4\%\) chance of being defective. A random sample of \(50\) components is selected for quality control. a) Find the probability of each event. (1) Exactly \(2\) components in the sample are defective. (2) At least \(4\) components in the sample are defective. (3) At most \(3\) components in the sample are defective. b) During one shift, \(100\) components have already been inspected and none were defective. Find the probability that the next component inspected is defective.

At a distribution center, \(15\%\) of packages are set aside because their addresses cannot be read automatically. During one shift, \(120\) packages are processed, and outcomes are treated as independent. Find the probability that at least \(12\) and at most \(24\) packages are set aside.

Determine whether each experiment can be described by a binomial model. Justify your decision. When appropriate, give the success probability \(p\), the number of trials \(n\), and the meaning of \(X\). a) A container holds \(40\) balls, \(8\) of which are red. Five balls are drawn with replacement. Each draw is recorded as red or not red. b) A basketball player has a \(75\%\) free-throw success rate and takes \(10\) free throws. The number of made shots is recorded. c) A box contains \(20\) batteries, \(2\) of which are defective. Two batteries are drawn without replacement and tested.

A gardener plants \(6\) seeds of a particular flower variety. Each seed has a germination probability of \(80\%\), and the germination outcomes are independent. a) Explain why the germination outcomes can be modeled as a sequence of \(6\) Bernoulli trials. b) Find the probability that exactly \(5\) of the \(6\) seeds germinate. c) Find the probability that at most \(1\) seed germinates.

A factory produces glass bottles. Historical data show that \(4\%\) of the bottles contain small air bubbles. A random sample of \(10\) bottles is selected for quality control. a) State the conditions under which this sample can be modeled using Bernoulli trials. For parts b) and c), assume those conditions hold. b) Find the probability that exactly \(1\) bottle in the sample contains an air bubble. c) Find the probability that at least \(2\) bottles in the sample contain air bubbles.

A spinner has \(4\) equal sections, and \(1\) section is red. The spinner is spun independently \(4\) times. a) Find the probability that the spinner lands on red exactly once. b) A person claims, “Since the probability of red on each spin is \(25\%\), the probability of getting red at least once in \(4\) spins is greater than \(90\%\).” Use a calculation to evaluate the claim.

A multiple-choice test has \(5\) questions. Each question has \(3\) answer choices, exactly one of which is correct. A student guesses independently on every question. Create a probability distribution table for the number \(X\) of questions answered correctly. Round each probability to four decimal places.

A spinner has \(3\) equal sections colored red, blue, and yellow. The spinner is spun independently \(6\) times. 1. Find the probability that the spinner lands on blue exactly \(2\) times. 2. Find the probability that the spinner lands on red at least \(5\) times.

A factory produces LED light bulbs. Each bulb is independently defective with probability \(0.05\). An inspector selects \(8\) bulbs and tests them. Find the probability that a) all \(8\) bulbs are nondefective, b) exactly \(2\) bulbs are defective, c) at least \(1\) bulb is defective.

An electronic component is defective with probability \(p=0.10\). For quality control, \(15\) components are randomly selected from a large shipment, with replacement. Find the probability that the sample contains no defective components or exactly \(1\) defective component.

A spinner has \(5\) equal sections. One section has a star and awards a prize. A player spins the spinner independently \(10\) times. Find the probability that the spinner lands on the prize section at least \(3\) times.

A large bag of gummy bears contains several colors, and the probability of selecting a red gummy bear is \(20\%\). A person makes \(15\) selections with replacement. Find the probability that at most \(2\) of the selected gummy bears are red.

Sarah and Tim play \(7\) table-tennis games. The player who wins at least \(4\) games wins the match. Sarah wins each game with probability \(0.55\), independently of the other games. Find the probability that Sarah wins the match.

Marc and Lucas play a series of \(10\) independent video games. Marc wins each game with probability \(0.40\). The overall winner is the player who wins more than half of the games. If each player wins \(5\) games, there is no overall winner. Find the probability that Marc is the overall winner.

A factory produces electronic components. Each component is independently defective with probability \(0.04\). A sample of \(15\) components is selected for quality control. Find the probability that a) exactly \(1\) component is defective, b) at least \(2\) components are defective, c) no components are defective.

An independent movie theater has \(13\) seats available for a special screening. Because each person with a reservation independently has a \(10\%\) probability of not attending, the theater accepts \(15\) reservations. Find the probability that more people attend than there are seats.

A small restaurant has \(10\) outdoor tables. Each evening reservation is independently canceled at the last minute with probability \(0.25\). The owner accepts \(12\) reservations for one evening. Find the probability that there are enough tables for everyone who arrives.

An automaker finds that \(8\%\) of a certain vehicle model have a minor paint defect. A random sample of \(30\) vehicles is inspected. Assume the sampled vehicles' defect statuses are independent. What number of vehicles with a paint defect has the greatest probability?

At a greenhouse, each seed of a certain flower variety germinates independently with probability \(0.75\). Fifteen seeds are planted. Which number or numbers of germinating seeds have the greatest probability?

A bag contains five balls labeled \(1\), \(2\), \(3\), \(4\), and \(5\). A ball is drawn at random, its number is recorded, and the ball is replaced. This process is repeated for a total of \(3\) draws. Find the probability that the ball labeled \(2\) is drawn exactly once.

A gardener plants \(12\) bulbs of a rare tulip variety. Each bulb sprouts with probability \(0.95\), independently of the others. Find the probability that exactly \(11\) of the \(12\) bulbs sprout.

A basketball player independently takes \(10\) free throws during practice and makes each free throw with probability \(0.70\). Find the probability of making exactly \(7\) free throws.

A multiple-choice test has \(15\) questions. Each question has four answer choices, exactly one of which is correct. A student guesses independently and randomly on every question. Find the probability that the student answers no questions correctly.

A shipping company reports that \(95\%\) of its packages arrive at their destination within \(24\) hours. A customer tracks \(20\) independent shipments. Find the probability that all \(20\) packages arrive within \(24\) hours.

A multiple-choice test has \(10\) questions. Each question has four answer choices, exactly one of which is correct. A student guesses independently and randomly on every question. a) Define a random variable \(X\), and identify its distribution and parameters. b) Find the probability that the student answers exactly half of the questions correctly. c) Find the expected number of correct answers.

An online store includes a coupon in each promotional package with probability \(0.2\), independently of the other packages. A customer receives \(12\) packages. Let \(X\) be the number that contain a coupon. a) Find the probability that exactly \(3\) packages contain a coupon. b) Find the probability that at least \(1\) package contains a coupon.

A spinner has \(8\) equal sectors numbered \(1\) through \(8\). a) Describe one way to classify a single spin as a Bernoulli trial. Give the success probability \(p\) and failure probability \(q\). b) Give another definition of success that produces a different probability \(p'\).

An electronics manufacturer reports that \(12\%\) of its microchips are defective. A random sample of \(4\) chips is selected from ongoing production. The random variable \(X\) is the number of defective chips in the sample. Determine the probability distribution of \(X\), rounding probabilities to four decimal places.

In a region, \(28\%\) of households have fiber-optic internet service. Five households are selected independently at random. The random variable \(Y\) is the number of sampled households with fiber-optic service. Determine the complete probability distribution of \(Y\), rounding to four decimal places.

An archer hits the center of a target with probability \(0.70\) on each shot and takes \(4\) shots. a) State the conditions under which this experiment can be modeled with a binomial distribution. b) Find the probabilities of exactly \(0\), \(1\), \(2\), \(3\), and \(4\) center hits.

Let \(X\sim\operatorname{Bin}(30,0.2)\). Find each probability. a) \(P(X\le 4)\) b) \(P(X>8)\) c) \(P(5\le X\le 10)\)

Let \(X\sim\operatorname{Bin}(80,0.35)\). Find each probability. a) \(P(X\le 20)\) b) \(P(22<X<32)\)

A tour-bus company has \(50\) seats available for a day trip. Each customer with a reservation independently has a \(10\%\) probability of not arriving, so the company accepts \(54\) reservations. Find the probability that every customer who arrives will have a seat.

A catering company estimates that each guest independently orders a vegetarian meal with probability \(0.15\). For an event with \(450\) guests, let \(X\) be the number of vegetarian orders. Find the probability that at least \(80\) guests order a vegetarian meal.

A binomial random variable \(X\) has parameters \(n=800\) and \(p=0.12\). Find \(P(80\le X\le 110)\).

A factory produces components that meet quality standards with probability \(p=0.85\), independently of one another. A random sample of \(n=150\) components is inspected. Let \(X\) be the number that meet the standards. Find the probability that between \(122\) and \(132\) components, inclusive, meet the standards.

A spinner has four equal sections labeled red, blue, green, and yellow. The spinner is spun \(40\) times independently. Let \(X\) be the number of times it lands on red. a) Find the probability that the spinner lands on red more than \(12\) times. b) Find the probability that it lands on red at least \(8\) times but no more than \(12\) times. c) Find the probability that it lands on red fewer than \(5\) times.

A sports club has a stadium section with \(1200\) seats. Because an average of \(15\%\) of season-ticket holders do not attend a game, the club sells \(1400\) season tickets. Assume that individual attendance is independent. Find the probability that more people attend than there are seats.

A spinner has two outcomes: win with probability \(p=0.3\), and lose with probability \(0.7\). The spinner is spun \(6\) times. a) Find the probability of exactly \(2\) wins. b) Briefly state the three main conditions needed to model this process as a binomial setting.

A bag contains \(4\) red balls, \(3\) green balls, and \(3\) yellow balls, as shown. A ball is drawn at random and replaced after each draw, for a total of six draws. Find the probability that a green ball is drawn exactly twice. Give the result as a percentage rounded to one decimal place.

The spinner shown has \(6\) equal sections: \(3\) yellow, \(2\) blue, and \(1\) red. It is spun independently \(6\) times. Find each probability. Round each percentage to two decimal places. a) The spinner lands on yellow exactly four times. b) The spinner lands on blue exactly two times.

The spinner shown has \(10\) equal sections and is spun independently \(8\) times. Find the probability that the spinner lands on red exactly \(3\) times. Give the answer as a decimal rounded to three decimal places.

The spinner shown has \(8\) equal sections and is spun independently \(10\) times. Find each probability, expressed as a percentage rounded to two decimal places. a) The spinner lands on red exactly four times. b) The spinner lands on yellow exactly two times.

The spinner shown has \(8\) equal sections colored yellow, blue, and red. Each spin is independent. Find the probability of each event. a) In \(5\) spins, the spinner lands on blue exactly \(2\) times. b) In \(8\) spins, the spinner lands on yellow exactly \(4\) times. c) In \(4\) spins, the spinner lands on red exactly once.

The spinner shown has \(5\) equal sections and is spun independently \(6\) times. Let \(X\) be the number of times the spinner lands on blue. Use the binomial probability formula to find \(P(X=3)\).

A fair six-sided number cube is rolled \(3\) times. Rolling a \(6\) is considered a success. The random variable \(X\) is the number of sixes rolled. a) Find \(P(X=k)\) for \(k\in\{0,1,2,3\}\). b) Verify that the probabilities add to \(1\).

At a glass factory, each drinking glass independently has a \(3\%\) chance of having small air bubbles. The glasses are packed in boxes of \(12\). a) Find the probability that a randomly selected box contains no glasses with air bubbles. b) An inspector checks \(6\) such boxes. What is the probability that at least \(5\) of the boxes contain only glasses without air bubbles?

Two basketball players compete in a free-throw challenge. Assume all shots are independent. Player A makes each free throw with probability \(0.90\) and takes \(4\) shots. Player B makes each free throw with probability \(0.60\) and takes \(6\) shots. Which event is more likely? Event 1: Player A makes at least \(3\) shots. Event 2: Player B makes exactly \(4\) shots.

In a game show, contestants choose between two strategies and guess every answer independently and randomly. Strategy A: Answer \(5\) multiple-choice questions, each with \(4\) choices and exactly one correct answer. To win, the contestant must answer at least \(3\) questions correctly. Strategy B: Answer \(10\) true-or-false questions. To win, the contestant must answer exactly \(5\) questions correctly. Which strategy has the greater probability of success?

A pharmaceutical company packages tablets in blister packs of \(20\). Suppose each tablet has the same probability \(p\) of being defective, independently of the others. The probability that a randomly selected blister pack contains at least one defective tablet is exactly \(5\%\). Find \(p\). Round your answer to four decimal places.

A spinner is divided into \(5\) equal sections, and exactly \(1\) section is red. The spinner is spun \(8\) times, and the spins are independent. a) Find the probability that the spinner lands on red exactly \(3\) times. b) Find the probability that the spinner lands on red at least \(2\) times. c) What is the minimum number of spins needed so that the probability of landing on red at least once is greater than \(90\%\)?

A multiple-choice vocabulary quiz has \(10\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A student guesses randomly on every question. a) Find the probability that the student answers exactly \(3\) questions correctly. b) Find the probability that the student answers at least \(1\) question correctly. c) What should the minimum passing score be so that the probability of passing by guessing alone is less than \(15\%\)?

A hostel has \(45\) beds. For one weekend, it accepts \(52\) reservations because historical data show that each guest with a reservation arrives with probability \(0.80\). Assume the guests’ arrival decisions are independent. a) What is the probability that more guests arrive than there are beds? b) The hostel wants the probability of overbooking—more than \(45\) guests arriving—to be at most \(10\%\). What is the maximum number of reservations it may accept?

An electronics manufacturer knows that \(5\%\) of its components are defective. A random sample of \(20\) components is selected. Let \(X\) be the number of defective components in the sample, and assume the situation can be modeled with independent Bernoulli trials. Find the probability of each event: a) Exactly \(1\) component is defective. b) At most \(2\) components are defective. c) At least \(1\) component is defective.

An archer hits the target on each shot with probability \(p=0.7\). The archer takes \(10\) independent shots, and \(X\) is the number of hits. Find each probability. a) At least \(8\) hits b) More than \(5\) but fewer than \(9\) hits

A spinner is divided into \(4\) equal sections colored red, blue, green, and yellow. The spinner is spun \(10\) times, and the spins are independent. Find the probability of each event: a) Red appears exactly \(2\) times. b) Red appears at most \(1\) time. c) Red appears at least \(2\) times. d) The spinner lands on a color other than red at least \(9\) times.

A fair six-sided die is rolled \(6\) times, and the rolls are independent. Find the probability of each event: a) Exactly one \(6\) is rolled. b) No \(6\) is rolled. c) At least two \(6\)s are rolled. d) A number other than \(6\) is rolled at most \(5\) times.

A spinner has \(5\) equal sections, exactly \(1\) of which is blue. The spinner is spun \(6\) times, the spins are independent, and landing on blue counts as a success. a) Find the probability of getting at least \(2\) successes. b) Find the probability that the first success occurs on the fourth spin. c) An event \(A\) has probability \(P(A)=\binom{6}{3}(0.2)^3(0.8)^3\). State the values of \(n\), \(k\), and \(p\), and describe event \(A\) in context.

Examine each probability expression. Determine whether it is a correct use of the binomial probability formula. For each valid expression, identify \(n\), \(p\), and \(k\), and describe a possible experiment and event. a) \(P(X=3)=\binom{10}{3}(0.15)^3(0.85)^7\) b) \(P(X=4)=\binom{12}{4}(0.20)^4(0.80)^6\) c) \(P(X=8)=\binom{10}{8}(0.20)^2(0.80)^8\)

A candy maker produces chocolates, and \(4\%\) of them are misshapen. A random sample of \(15\) chocolates is selected. Let \(X\) be the number of misshapen chocolates, and assume the outcomes are independent. a) Explain why \(P(X=2)=\binom{15}{2}(0.04)^2(0.96)^{13}\) correctly represents the probability of selecting exactly \(2\) misshapen chocolates. b) An employee claims that \((0.96)^{15}+15(0.04)(0.96)^{14}\) gives the probability that at most \(1\) chocolate is misshapen. Evaluate the claim using the binomial probability formula.

A multiple-choice test has \(12\) questions. Each question has four answer choices, exactly one of which is correct. A student guesses randomly and independently on every question. Let \(X\) be the number of correct answers. a) Find the probability that the student answers at least \(5\) questions correctly. b) Find the smallest integer \(k\) for which the probability of getting at least \(k\) correct answers by guessing is less than \(10\%\).

A container holds \(20\) marbles. Some are blue, and the rest are yellow. A marble is drawn \(8\) times with replacement. The probability of drawing at most \(2\) blue marbles is approximately \(67.85\%\). How many blue marbles are in the container?

A shipment contains \(25\) electronic components. For quality control, a component is selected, checked, and returned to the shipment before the next selection. This is repeated \(10\) times. The probability of selecting at least \(1\) defective component is approximately \(89.26\%\). How many defective components are in the shipment?

A binomial random variable \(X\) has parameters \(n=20\) and \(p=0.30\). Find each probability. a) \(P(X=6)\) b) \(P(X\le 4)\) c) \(P(X>7)\) d) \(P(5\le X\le 9)\)

The lifetime \(X\), in hours, of an LED bulb is normally distributed with mean \(\mu=15{,}000\) and standard deviation \(\sigma=1200\). 1. Find the probability that a randomly selected bulb lasts less than \(13{,}500\) hours. 2. A wholesaler samples \(200\) bulbs. Find the probability that at most \(30\) bulbs in the sample last less than \(13{,}500\) hours. Model the count with a binomial distribution.

A hot-drink machine dispenses coffee with volume \(X\), in milliliters, normally distributed with mean \(\mu=200\) and standard deviation \(\sigma=5\). a) Find the probability that a cup contains less than \(195\,\text{mL}\). b) For a meeting, \(50\) cups are dispensed independently. Find the probability that at most \(10\) cups contain less than \(195\,\text{mL}\).

A factory produces precision bolts. Their diameters \(X\), in millimeters, are normally distributed with mean \(\mu=10.00\) and standard deviation \(\sigma=0.05\). A bolt fits correctly if its diameter is between \(9.95\,\text{mm}\) and \(10.05\,\text{mm}\). a) Find the proportion of bolts that fit correctly. b) A shipment contains \(200\) bolts. Find the probability that at least \(140\) fit correctly.

A manufacturer states that the lifetime \(X\), in hours, of an LED model is normally distributed with mean \(\mu=1500\) and standard deviation \(\sigma=60\). a) Find the probability that a randomly selected LED lasts at least \(1410\) hours. b) An industrial facility installs \(40\) LEDs. Find the probability that at least \(38\) of them last at least \(1410\) hours.

A manufacturer states that the lengths of its steel pins are normally distributed with mean \(\mu=120\,\text{mm}\) and standard deviation \(\sigma=0.5\,\text{mm}\). a) Describe how the manufacturer could have estimated \(\mu\) and \(\sigma\) from a sample. b) A customer buys a package of \(150\) pins. A pin meets specifications when its length is between \(119.5\,\text{mm}\) and \(120.5\,\text{mm}\). Find the probability that at least \(65\%\) of the pins in the package meet specifications.

A coffee company fills bags with ground coffee. The fill weight \(W\), in grams, is normally distributed with \(\mu=500\) and \(\sigma=4\). a) Find the probability that a randomly selected bag contains more than \(505\,\text{g}\). b) A grocery store receives \(400\) bags. Find the probability that at most \(10\%\) of the bags contain more than \(505\,\text{g}\). Assume the bag weights are independent.

At a bottling plant, the fill amount \(X\) in a \(750\,\text{mL}\) orange juice bottle is normally distributed with mean \(\mu=754\,\text{mL}\) and standard deviation \(\sigma=4\,\text{mL}\). a) Find the probability that a randomly selected bottle contains less than \(750\,\text{mL}\). b) A case contains \(12\) bottles. Find the probability that at most one bottle contains less than \(750\,\text{mL}\). Assume the fill amounts are independent. c) What is the minimum number of bottles that must be inspected so that the probability of finding at least one bottle containing less than \(750\,\text{mL}\) is at least \(95\%\)?

The diameter \(D\) of a steel pin is normally distributed with mean \(\mu=20.00\,\text{mm}\) and standard deviation \(\sigma=0.05\,\text{mm}\). A pin is rejected if its diameter differs from the mean by more than \(0.10\,\text{mm}\). a) Use the Empirical Rule, without a calculator, to explain why about \(5\%\) of the pins are rejected. b) A box contains \(50\) independently produced pins. Find the probability that exactly two pins are rejected. c) A normal approximation may be used for a binomial distribution when the large-counts conditions \(np\ge10\) and \(n(1-p)\ge10\) hold. Find the minimum value of \(n\) that meets both conditions, and verify the conditions for \(n=200\).

The fill weight \(X\) of a coffee bag is approximately normally distributed with mean \(\mu=500\,\text{g}\) and standard deviation \(\sigma=5\,\text{g}\). The bags are classified as follows: <table> <tr> <td>Category</td> <td>Underweight</td> <td>Standard</td> <td>Overweight</td> </tr> <tr> <td>Weight \(m\) in \(\text{g}\)</td> <td>\(m<490\)</td> <td>\(490\le m<510\)</td> <td>\(m\ge510\)</td> </tr> </table> a) Use the Empirical Rule to estimate the probability that a randomly selected bag is classified as Standard and the probability that it is classified as Overweight. b) Find the minimum number of independently selected bags needed so that the probability of including at least one Underweight bag is at least \(95\%\). Use the Empirical Rule estimate from part a).

An orange juice filling machine produces bottles whose fill amount \(X\), in milliliters, is normally distributed with mean \(\mu=500\) and standard deviation \(\sigma=4\). a) Find the probability that a randomly selected bottle contains between \(495\,\text{mL}\) and \(505\,\text{mL}\). b) Find the proportion of bottles containing less than \(492\,\text{mL}\). c) Twelve bottles are selected independently for a case. Find the probability that at most one bottle in the case contains less than \(492\,\text{mL}\).

The battery life \(X\), in hours, of a smartphone under heavy use is normally distributed with mean \(\mu=14\) and standard deviation \(\sigma=1.5\). a) Find the probability that a randomly selected phone has a battery life between \(12\) and \(15\) hours. b) Find the percentage of phones with a battery life longer than \(16\) hours. c) A retailer has \(50\) independently selected phones. Find the probability that at least \(5\) have a battery life longer than \(16\) hours.

A multiple-choice test has \(6\) questions. Each question has \(5\) answer choices, exactly one of which is correct. A student guesses randomly on every question. Describe an event whose probability is represented by each expression. a) \(\left(\frac{4}{5}\right)^6\) b) \(1-\left(\frac{4}{5}\right)^6\) c) \(6\cdot\frac{1}{5}\cdot\left(\frac{4}{5}\right)^5\) d) \(1-\left[\left(\frac{4}{5}\right)^6+6\cdot\frac{1}{5}\cdot\left(\frac{4}{5}\right)^5\right]\)

For a group of \(15\) people, assume that each person’s day of birth is independently and equally likely to be any of the \(7\) days of the week. a) Find the probability that exactly \(2\) people were born on a Sunday. b) Find the probability that at least \(1\) person was born on a Saturday. c) Find the probability that more than \(13\) people were born on a weekday from Monday through Friday.

A digital memory chip stores data in blocks of \(20\) bits, where each bit is either \(0\) or \(1\). a) How many different \(20\)-bit blocks contain exactly \(2\) ones? b) During transmission, each bit independently has a constant error probability of \(p=0.01\). Find the probability that a \(20\)-bit block has at most \(1\) error. c) State the conditions under which the number of errors in a block can be modeled with a binomial distribution.

A logistics company labels storage bins with codes. Each code contains either one or two uppercase letters from \(\mathrm{A}\) through \(\mathrm{Z}\), followed by a two-digit number from \(01\) through \(99\). Assume all valid codes are equally likely. A code is called special when its first letter is \(\mathrm{A}\). 1. Find the total number of possible codes. 2. Find the probability \(p\) that a randomly generated code is special. 3. Twenty codes are independently selected with replacement. Find the probability that exactly \(3\) are special.

An archive assigns each document a code consisting of one letter from \(\text{A}\), \(\text{B}\), \(\text{C}\), \(\text{D}\), or \(\text{E}\), followed by a three-digit string from \(000\) through \(999\). A code is called a priority code when its number begins with \(9\). 1. Show that the probability \(p\) that a randomly generated code is a priority code is \(0.1\). 2. Ten codes are generated independently. Find the probability that more than \(2\) are priority codes.

Five friends—Ava, Ben, Clara, David, and Elias—sit in a random order in a row of five movie theater seats. a) Describe a model using draws from a bag for this random process. Find the probability \(p\) that Ava and Ben sit next to each other. b) The five friends go to the movies on \(10\) different nights and choose a new random seating order each time. Find the probability that Ava and Ben sit next to each other on exactly \(4\) nights.

An online retailer knows that \(15\%\) of shipped packages are returned by customers. A random sample of \(25\) packages is selected, and the return outcomes can be modeled as independent Bernoulli trials. a) Find the probability that among the \(25\) packages: (1) exactly \(3\) are returned. (2) none are returned. (3) at least \(2\) are returned. b) During a promotion, the return rate is expected to increase to \(20\%\). Find the expected number of returns in a sample of \(25\) packages and compare it with the expected number when the return rate is \(15\%\).

A shipping company finds that each package independently has a \(15\%\) chance of requiring special handling because of its dimensions. A sample of \(20\) packages is inspected. a) Find the number of different sequences of special handling \((S)\) and normal handling \((N)\) that contain exactly \(4\) packages requiring special handling. b) Find the probability that exactly \(4\) packages require special handling. c) Find the probability that at most \(2\) packages require special handling.

A fair six-sided die is rolled \(6\) times. Treat rolling a \(6\) as a success and any other result as a failure. Consider these two exact sequences: Sequence A: \((6, 6, 6, 6, 6, 6)\) Sequence B: \((6, \text{not }6, 6, \text{not }6, \text{not }6, 6)\) a) Without calculating, decide which sequence is more likely. Explain your reasoning. b) Calculate \(P(A)\) and \(P(B)\). c) Find the probability that exactly three of the six rolls show a \(6\). Explain how this probability differs from \(P(B)\).

A container holds \(40\) components, \(6\) of which are defective. Five components are selected one at a time. Determine whether the experiment is binomial when the components are selected a) with replacement; b) without replacement. For the case that is binomial, find the probability of selecting exactly one defective component.

A machine produces bolts that independently meet a quality standard with probability \(0.85\). Six bolts are selected and inspected in order. Find the probability of each event. a) Only the third bolt fails to meet the standard. b) All six bolts meet the standard. c) Only the first four bolts meet the standard. d) Exactly five of the six bolts meet the standard.

In a manufacturing process, each component independently has a \(5\%\) chance of being defective. A sample of \(100\) components is selected. Let \(Y\) be the number of defective components in the sample. Describe each expression in context and find its value. a) \(\sum_{k=0}^{10}\binom{100}{k}(0.05)^k(0.95)^{100-k}\) b) \(1-\sum_{k=0}^{2}\binom{100}{k}(0.05)^k(0.95)^{100-k}\) c) \(\sum_{k=4}^{7}\binom{100}{k}(0.05)^k(0.95)^{100-k}\)

A random variable \(X\) has a binomial distribution, and its cumulative distribution function is \(F(k)=P(X\le k)\). For integers \(a\) and \(b\) in \(\{0,1,\ldots,n\}\) with \(a<b\), explain why \(P(a<X\le b)=F(b)-F(a)\).

Let \(X\sim\operatorname{Bin}(150,0.6)\). Write each probability using summation notation, and find its value. Round to four decimal places. a) \(P(X<85)\) b) \(P(X\ge 100)\) c) \(P(80\le X<110)\)

Let \(X\sim\operatorname{Bin}(80,0.25)\). a) List all possible values of \(X\) that satisfy \(|X-20|\le 3\). b) Find each probability. (1) \(P(|X-20|\le 3)\) (2) \(P(|X-20|>3)\) (3) \(P(|X-18|<2)\)

An archer hits the bullseye with probability \(0.70\) on each shot, independently of the other shots. The archer takes \(8\) shots. Find the probability that the archer: a) hits the bullseye exactly \(6\) times. b) hits the bullseye at least \(7\) times. c) hits the bullseye at most \(5\) times.

For each expression, describe the event it represents and find its probability. a) For \(X\sim\operatorname{Bin}(8,0.25)\), evaluate \(\sum_{i=3}^{5}P(X=i)\). b) For \(Y\sim\operatorname{Bin}(12,0.5)\), evaluate \(\sum_{i=4}^{6}P(Y=i)\).

A spinner has \(8\) equal sections labeled \(1\) through \(8\). The spinner is spun \(10\) times, and the spins are independent. Find the probability of each event. 1. A prime number appears exactly \(4\) times. 2. A prime number appears at most \(3\) times. 3. A number greater than \(6\) appears at least \(8\) times. 4. A prime number appears on only the first \(3\) spins, and no prime number appears on any later spin.

A marble is drawn repeatedly from a bag with replacement. The probability of drawing a red marble is \(p\). Interpret each expression by stating the number of draws \(n\), the red-marble probability \(p\), and one event \(E\) represented by the expression. 1. \(\binom{15}{4}(0.2)^4(0.8)^{11}\) 2. \(\sum_{k=0}^{2}\binom{12}{k}(0.4)^k(0.6)^{12-k}\) 3. \((0.3)^3(0.7)^7\) 4. \(1-\sum_{k=0}^{13}\binom{20}{k}(0.9)^k(0.1)^{20-k}\)

A multiple-choice test has \(20\) questions. Each question has five answer choices, exactly one of which is correct. A student guesses independently and randomly on every question. Determine the most likely number \(k\) of correct answers and the least likely number \(k\) of correct answers. Justify the least likely value by comparing the probabilities at the endpoints.

A gardener plants \(20\) seeds of a certain flower variety. Each seed germinates independently with probability \(0.85\). Let \(X\) be the number of seeds that germinate. Find the probability of each event. a) Exactly \(18\) seeds germinate. b) At most \(16\) seeds germinate. c) At least \(15\) seeds germinate. d) More than \(16\) but fewer than \(20\) seeds germinate.

Let \(X\) be a binomial random variable with \(n=30\) and \(p=0.6\). Use a calculator or spreadsheet to determine: a) every value \(k\in\{0, 1, \ldots, 30\}\) for which \(P(X=k)\ge0.1\); b) every value \(k\in\{0, 1, \ldots, 30\}\) for which \(P(X=k)<0.05\).

A binomial random variable \(X\) has parameters \(n=100\) and \(p=0.25\). Use appropriate technology to find: a) the smallest nonnegative integer \(k\) such that \(P(X\le k)\ge 0.95\); b) the smallest nonnegative integer \(k\) such that \(P(X>k)\le 0.01\).

Consider a binomial setting with \(n\) trials and success probability \(p=0.5\), such as repeated flips of a fair coin. Let \(X\) be the number of successes. a) Use \(P(X=k)=\binom nkp^k(1-p)^{n-k}\) to prove that \(P(X=k)=P(X=n-k)\) for every \(k\in\{0, 1, \ldots, n\}\). b) For \(n=6\), calculate \(P(X\ge4)\), and use the symmetry from part a) to show that it equals \(P(X\le2)\).

A manufacturer produces LED light strips, each containing \(20\) individual LEDs. Each LED is defective with probability \(0.02\), independently of the others. Assume the light strips are also independent. a) Find the probability that a randomly selected light strip has no defective LEDs. b) A retailer receives \(15\) such light strips. Find the probability that exactly \(12\) strips are completely functional.

A glass-bottle manufacturer reports that each bottle independently has a \(4\%\) chance of having small inclusions and being classified as a second. A sample of \(200\) bottles is selected. Let \(X\) be the number of bottles with inclusions. Find the probability of each event. a) Exactly \(8\) bottles have inclusions. b) At most \(10\) bottles have inclusions. c) From \(5\) through \(12\) bottles, inclusive, have inclusions.

A multiple-choice test has \(30\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A student answers every question by random guessing. Let \(X\) be the number of correct answers. Find each probability. a) More than \(10\) correct answers b) At least \(5\) but fewer than \(10\) correct answers

A seed company reports that a certain variety has a germination probability of \(0.85\). A gardener plants \(30\) seeds. For parts 1 and 2, assume the germination outcomes are independent and each seed has the same germination probability. 1. Find the probability that at most \(22\) seeds germinate. 2. Find the probability that at least \(28\) seeds germinate. 3. Explain the conditions under which a binomial model is appropriate in this situation.

A manufacturer reports that \(4\%\) of its LED bulbs have minor scratches on the housing. A random sample of \(250\) bulbs is inspected, and the inspections are treated as independent. Let \(X\) be the number of scratched bulbs in the sample. Find each probability. (1) At most \(5\) scratched bulbs (2) More than \(12\) scratched bulbs (3) At least \(8\) but at most \(15\) scratched bulbs

A study reports that each resident in a city independently has a \(15\%\) chance of regularly using a particular car-sharing service. A sample of \(80\) residents is selected. Find the probability that among those surveyed: (1) fewer than \(10\) regularly use the service. (2) at least \(15\) regularly use the service. (3) exactly \(12\) regularly use the service.

An LED manufacturer reports that \(2.5\%\) of its bulbs are defective. A home-improvement store receives a shipment of \(80\) bulbs. Assume the number of defective bulbs is binomially distributed. Find the probability of each event. Event \(A\): Exactly \(2\) bulbs are defective. Event \(B\): More than \(5\%\) of the bulbs in the shipment are defective.

In a quality-control process for glass bottles, \(15\%\) of the bottles have small air bubbles. A random sample of \(60\) bottles is selected. Let \(X\) be the number of bottles with air bubbles, and assume \(X\) is binomially distributed. Find the probability of each event. Event \(C\): At most \(5\) bottles have air bubbles. Event \(D\): From \(8\) through \(12\) bottles, inclusive, have air bubbles.

A gardener buys a packet of \(15\) flower seeds. The manufacturer reports a germination probability of \(0.90\), and the seeds are assumed to germinate independently. a) Find the probability that at least \(14\) of the \(15\) seeds germinate. b) Another gardener needs exactly \(10\) plants and buys \(11\) seeds, saying, “Buying \(10\%\) extra guarantees that I will get \(10\) plants.” Evaluate this statement using a probability calculation.

An LED manufacturer finds that \(5\%\) of its bulbs are defective, independently of one another. a) An inspector selects a sample of \(50\) bulbs. Find the probability that at most \(3\) bulbs are defective. b) An employee claims, “In a sample of \(50\), exactly \(5\) defective bulbs is more likely than no defective bulbs.” Determine whether the claim is correct.

A biathlete shoots at a series of \(5\) targets in practice. Her average hit probability per shot is \(0.80\). a) State the theoretical conditions needed to model the number of hits with a binomial distribution having \(n=5\) and \(p=0.80\). b) Explain why independence of the shots might be questionable in an actual competition. c) Suppose a miss lowers her probability of hitting the immediately following target to \(0.60\). Which condition of a binomial model is violated?

A company manufactures electronic components, and \(15\%\) receive a special quality seal. A random sample of \(12\) components is selected without replacement from a very large production lot. Find the probability that at least \(4\) selected components have the quality seal. Also explain the condition under which a binomial distribution is a reasonable approximation for this sample.

A spinner has blue, red, and green sections. The blue section has a central angle of \(108^\circ\). The spinner is spun \(20\) times, the spins are independent, and landing on blue counts as a success. a) State the parameters \(n\) and \(p\) of the binomial distribution and explain what they represent. b) Find the probability that the spinner never lands on blue. c) Find the probability of at most \(1\) blue result.

A basketball player makes a free throw with probability \(0.75\), independently of the other shots. The player takes \(4\) free throws. a) Match each event with the corresponding probability expression, and explain your reasoning. Events: \(A\): Exactly \(2\) shots are made. \(B\): All \(4\) shots are made. \(C\): The first made shot occurs on the fourth attempt. \(D\): The first \(2\) shots are missed, and the last \(2\) are made. \(E\): At least \(3\) shots are made. Expressions: (1) \((0.75)^4\) (2) \((0.25)^3(0.75)\) (3) \((0.25)^2(0.75)^2\) (4) \(\binom{4}{2}(0.75)^2(0.25)^2\) (5) \(\sum_{k=3}^{4}\binom{4}{k}(0.75)^k(0.25)^{4-k}\) b) Find the probability of Event \(E\).

A factory produces electronic components, and \(4\%\) are defective. Ten components are selected in order, and the selections can be modeled as independent Bernoulli trials. a) Find the probability that the first defective component occurs in the third position. b) Find the probability that at most \(1\) component in the sample is defective. c) Explain mathematically why the events “only the first component is defective” and “only the last component is defective” have the same probability.

A seed company states that a certain variety of tomato seed has a germination rate of \(85\%\). A gardener plants \(30\) seeds, and the germination outcomes are independent. a) Find the most likely number of seeds that germinate. b) Find the probability that at least \(24\) seeds germinate.

A battery manufacturer reports that \(94\%\) of its batteries last more than \(1000\) charge cycles. A random sample of \(50\) batteries is tested. Let \(X\) be the number that meet this standard. a) Assuming a binomial model, find the probability of each event. (1) Exactly \(48\) batteries meet the standard. (2) At most \(45\) batteries meet the standard. (3) At least \(46\), but fewer than \(50\), batteries meet the standard. b) State two conditions needed for a binomial model to be appropriate in this context.

At a tour company's information booth, \(20\%\) of visitors are interested in an international trip. During one hour, \(80\) visitors stop at the booth. Assume their interests are independent, and let \(X\) be the number interested in an international trip. a) What is the minimum number of international-travel brochures the company should have so that the brochures are sufficient for all interested visitors with probability at least \(95\%\)? b) Suppose the company has \(18\) brochures. Find the probability that at least one brochure remains at the end of the hour.

A home improvement store runs a promotion in which each customer rolls a fair six-sided die at checkout. A customer who rolls a \(6\) receives an instant-discount coupon. The die is rolled \(150\) times during the morning. a) What is the minimum number of coupons the store should prepare so that there are enough for all winners with probability at least \(90\%\)? b) The manager prepares \(30\) coupons. Find the probability that more customers win than there are coupons available.

A small movie theater has \(45\) seats for a documentary premiere. Based on past experience, each person with a reservation independently has a \(10\%\) chance of not attending. The theater accepts \(50\) reservations. Find the probability that every person who attends will have a seat.

A multiple-choice test has \(5\) answer choices for each question, exactly one of which is correct. A test taker guesses independently and randomly on every question. Find the minimum number of questions needed so that the probability of answering at least \(1\) question correctly is: a) at least \(80\%\). b) at least \(95\%\).

An egg producer reports that at most \(2\%\) of the eggs in a shipment have cracked shells. A grocery store inspects a carton of \(20\) eggs and rejects the shipment if more than \(1\) egg in the sample is cracked. a) Assuming the true cracked-shell rate is exactly \(2\%\), find the probability that the shipment is rejected. Assume a binomial model is appropriate. b) Find the rejection probability if the true cracked-shell rate is \(1\%\).

At a garden center, a certain type of seed germinates independently with probability \(0.90\). For each expression, describe a suitable random experiment and an event in context whose probability is represented by the expression. a) \(\binom{20}{18}(0.9)^{18}(0.1)^2\) b) \(1-[(0.1)^{10}+10(0.9)(0.1)^9]\) c) \(1-\sum_{k=0}^{16}\binom{20}{k}(0.9)^k(0.1)^{20-k}\)

An electronics manufacturer finds that each microchip independently has a \(5\%\) chance of being defective. The chips are shipped in packages of \(60\). a) Find the probability that more than \(10\%\) of the microchips in a randomly selected package are defective. b) During quality control, the chips in a package are tested one at a time. Describe an event in context whose probability is represented by \(\sum_{k=0}^{4}(0.95)^k(0.05)\). c) What minimum proportion of the manufacturer's microchips must be nondefective so that a sample of \(30\) chips contains no defective chips with probability at least \(80\%\)?

In an online game, each treasure chest independently has a \(12\%\) probability of containing a rare item. A player opens \(40\) chests. a) Find the probability that the player receives at most \(3\) rare items. b) Describe an event in context whose probability is \(1-(0.88)^{15}\). c) What is the minimum number of treasure chests a player must open so that the probability of receiving at least \(1\) rare item is at least \(99\%\)?

A manufacturer packages glass figurines in sets of \(5\). Historical data show that \(40\%\) of the figurines have minor flaws. Assume that flaws occur independently and that the sets are assembled independently. A retailer receives \(20\) sets. Find the probability that at most \(2\) of the sets contain only flawless figurines.

A gardener knows that \(80\%\) of a certain type of flower seed germinates, and seed outcomes are treated as independent. a) The gardener plants \(450\) seeds in a garden bed. How many seedlings should be expected? b) For a small pot, the gardener plants \(20\) seeds. Find the probability that exactly \(18\) germinate. c) Find the probability that at least \(19\) of the \(20\) seeds germinate.

A city survey estimates that \(35\%\) of eligible voters support a local ballot proposal. A random sample of \(150\) eligible voters is surveyed. The sample is small relative to the city’s voter population, so responses are modeled as approximately independent. Let \(X\) be the number who support the proposal. a) Find the probability that at least \(60\) people in the sample support the proposal. b) Find the probability that between \(45\) and \(60\) people, inclusive, support the proposal.

At a greenhouse, seeds of a certain flower variety germinate independently with probability \(0.92\). A random sample of \(120\) seeds is planted under identical conditions. Let \(X\) be the number of seeds that germinate. a) Find \(P(X=110)\). b) Find the number of germinating seeds for which \(P(X=k)\) is greatest.

A diagnostic screening test produces a false-positive result for \(2\%\) of healthy people. In a study, \(500\) healthy people are tested independently. Let \(X\) be the number of false-positive results. a) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). Explain the meaning of the mean in this context. b) Find the probability that \(X\) is within one standard deviation of its mean.

A manufacturer knows that \(10\%\) of its glass marbles contain tiny air bubbles and are classified as seconds. A package contains \(300\) randomly selected marbles, and outcomes are treated as independent. Let \(Y\) be the number of marbles with air bubbles. a) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(Y\). b) Find the probability that \(Y\) differs from its mean by at most two standard deviations.

The random variable \(Y\) has a binomial distribution with \(n=25\) and \(p=0.5\). a) Without calculating individual probabilities, justify why the probability distribution of \(Y\) has two adjacent values with the same maximum probability. b) State the two values and explain how the symmetry of the distribution supports your answer.

The random variable \(X\) has a binomial distribution with \(n=100\) and success probability \(p\). Find all values of \(p\) with \(0<p<1\) for which \(P(X=45)<P(X=55)\).

A random variable \(Y\) has a binomial distribution with \(n=3\) and \(p\in(0, 1)\). Find all values of \(p\) for which the probability of no successes is at least as great as the probability of exactly one success.

A binomial random variable \(X\) has parameters \(n=80\) and \(p=0.3\). Find the narrowest interval of the form \([\mu-k,\mu+k]\), symmetric about the mean, that contains at least \(95\%\) of the probability. Assume \(k\) is a nonnegative integer.

A binomial random variable \(Y\) has parameters \(n=150\) and \(p=0.2\). Find the narrowest interval of the form \([\mu-c,\mu+c]\), symmetric about the mean, that contains at least \(90\%\) of the probability. Assume \(c\) is a nonnegative integer.

At a customer service center, each call independently has a \(15\%\) chance of not being answered within the first \(20\) seconds. The center introduces a new staffing tool and evaluates it by selecting \(120\) calls and counting the calls that were not answered within \(20\) seconds. The tool is considered successful if the number of delayed calls is more than one standard deviation below the original expected number. Find the probability that the tool is classified as unsuccessful even though the actual delayed-call rate has decreased to \(8\%\).

An agricultural researcher studies the germination of a seed variety. In \(150\) trials, each using \(5\) seeds, the number \(k\) of seeds that germinated was recorded. <table> <tr><td>Number germinated \(k\)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td><td>\(5\)</td></tr> <tr><td>Frequency \(n_k\)</td><td>\(4\)</td><td>\(21\)</td><td>\(50\)</td><td>\(48\)</td><td>\(22\)</td><td>\(5\)</td></tr> </table> 1. Find the mean number \(\bar{k}\) of germinated seeds per trial. 2. Assume the number germinated follows \(\operatorname{Bin}(5,p)\). Estimate \(p\) by setting the observed mean equal to the binomial mean. 3. Under the fitted model, find the probability that exactly \(2\) seeds germinate. Compare it with the observed relative frequency.

An electronics manufacturer tests components in groups of \(3\). For \(200\) groups, the number \(k\) of defective components per group was recorded. <table> <tr><td>Number defective \(k\)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td></tr> <tr><td>Number of groups \(n_k\)</td><td>\(130\)</td><td>\(55\)</td><td>\(12\)</td><td>\(3\)</td></tr> </table> 1. Find the overall proportion of defective components in the sample. Use it as an estimate of the defect probability \(p\) for one component. 2. Assume the number of defects per group follows \(\operatorname{Bin}(3,p)\). Using the estimate from part 1, find the probability that a group contains at least \(1\) defective component. 3. Compare the model probability with the observed relative frequency of groups containing at least \(1\) defect.

A binomial random variable \(X\) has success probability \(p=0.5\) and expected value \(E(X)=6\). a) Determine the number of trials \(n\). b) Calculate \(P(X=4)\) and \(P(X=8)\), rounded to four decimal places. c) Use the binomial probability formula \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\) to explain why the histogram of any binomial distribution with \(p=0.5\) is symmetric about \(k=\frac{n}{2}\).

Let \(X\sim\operatorname{Bin}(21,0.5)\). a) Without using a table or a calculator command for cumulative probabilities, explain why \(P(X\le 10)=0.5\). Use the symmetry of the distribution. b) Calculate \(P(X=10)\), and then use it to find \(P(X<10)\). Round each result to four decimal places.

A manufacturing facility inspects a sample of \(500\) electronic components and finds \(15\) defective components. 1. Find the relative frequency of defective components. Identify the probability approach used when this relative frequency is treated as an estimate of the defect probability. 2. For planning purposes, a model assumes a constant defect probability of \(p=0.02\) and independent outcomes for \(20\) randomly selected components. Find the probability that exactly \(1\) component is defective. Identify the probability approach used in this model-based calculation.

Decide whether each situation is a binomial experiment. If it is, give \(n\) and \(p\). a) A fair six-sided die is rolled \(20\) times. On each roll, record whether the result is prime. b) A spinner has \(10\) equal sectors numbered \(1\) through \(10\). It is spun \(15\) times, and the exact number shown on every spin is recorded.

A spinner has \(12\) equal sections labeled \(1\) through \(12\). The spinner is spun \(25\) times, and the spins are independent. a) Find the probability that (1) a multiple of \(3\) occurs at least \(5\) times, (2) the number \(12\) occurs more than \(2\) times, (3) a prime number occurs at most \(10\) times. b) The spinner has been spun \(25\) times without landing on \(12\). Find the probability that it lands on \(12\) on the \(26\)th spin.

A manufacturer reports that each electronic component independently has a \(4\%\) chance of having a particular manufacturing defect. A technician selects a random sample of \(n\) components. What is the minimum sample size needed so that the probability of finding at least \(1\) component with this defect is at least \(95\%\)?

In a city, \(60\%\) of households have fiber-optic internet service. A random sample of \(100\) households is selected. The sample is small relative to the city’s household population, so households are modeled as approximately independent. Let \(X\) be the number in the sample that have fiber-optic service. 1. Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). 2. Find the probability that \(X\) is within one standard deviation of its mean.

A computer-chip manufacturer finds that \(8\%\) of its chips are defective. Let \(X_n\) be the number of defective chips in a sample of size \(n\), so \(X_n\sim\operatorname{Bin}(n,0.08)\). a) A sample of \(20\) chips is inspected. Find the probability of each event. \(E_1\): None of the chips are defective. \(E_2\): At most \(2\) chips are defective. \(E_3\): Exactly \(17\) or \(18\) chips are nondefective. b) Find the minimum value of \(n\) needed so that the probability of finding at least \(1\) defective chip is at least \(99\%\).

A gardener plants \(12\) seeds of a rare flower. Each seed germinates with probability \(0.85\), independently of the others. Find the probability of each event. (1) Exactly \(9\) seeds germinate. (2) At least \(11\) seeds germinate. (3) Only the first three planted seeds fail to germinate, while all remaining seeds germinate.

A quality-control sample contains \(15\) LED bulbs. Each bulb is independently defective with probability \(0.04\). Find the probability that: (1) exactly one bulb is defective; (2) more than two bulbs are defective; (3) only the last inspected bulb is defective.

In a clinical study, each patient responds positively to a treatment with probability \(0.4\). Fifty patients are treated, and outcomes are treated as independent. Let \(X\) be the number of positive responses. Find the probability that \(X\) differs from its mean by at most \(4\).

A fair coin is flipped independently \(8\) times. Let \(H\) represent heads and \(T\) represent tails. Consider these two sequences: Sequence 1: \((T, T, T, T, T, T, T, T)\) Sequence 2: \((H, T, T, H, T, H, H, T)\) a) Find the probability of each sequence. b) Many people think Sequence 2 is more likely because it appears more mixed. Explain why that intuition is mathematically incorrect. c) Let \(X\) be the number of tails in the \(8\) flips. Find \(P(X=4)\), and use it to explain why the event “exactly four tails” is much more likely than “eight tails.”

A basketball player independently takes \(4\) free throws and makes each free throw with probability \(0.80\). Let \(M\) represent a make and \(F\) represent a miss. a) Find the probabilities of these sequences: \(S_1=(M, M, M, M)\) \(S_2=(M, F, M, M)\) b) Evaluate the statement: “In a binomial setting, every sequence with the same number of successes has the same probability, regardless of the positions of the successes.” Justify your answer using an example with three makes. c) Find the probability that the player makes exactly three free throws.

A gardener buys \(20\) seeds of a rare plant. The seeds germinate independently, but the germination probability \(p\) for each seed under the storage conditions is unknown. Find the minimum threshold for \(p\) that makes the probability that at least one seed germinates at least \(95\%\). Give an exact expression and a decimal approximation.

A manufacturer tests a random sample of \(50\) microchips from a new production line. Assume chip defects are independent, and let \(p\) be the unknown defect rate. Find the minimum threshold for \(p\) that makes the probability of finding at least one defective chip in the sample at least \(90\%\). Give an exact expression and a decimal approximation.

Decide whether a binomial model is appropriate for each situation. a) In a large city survey, \(100\) people are randomly selected and asked whether they own an electric vehicle. About \(4\%\) of the city's residents own one. b) An archer takes \(12\) practice shots. As the session continues, fatigue lowers the archer's concentration. c) A spinner has three equal sectors colored blue, yellow, and red. It is spun \(20\) times, and each spin is classified as yellow or not yellow.

A chip manufacturer reports that \(5\%\) of its computer chips are defective. A distributor tests \(400\) chips from a shipment, and outcomes are treated as independent. Let \(X\) be the number of defective chips. a) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). b) Find the probability that \(X\) lies in the interval \([\mu-\sigma,\mu+\sigma]\).

In a small town, \(64\%\) of eligible voters support a local initiative. A random sample of \(150\) eligible voters is surveyed, and responses are treated as independent. Let \(X\) be the number who support the initiative. a) Find the mean \(\mu\) and standard deviation \(\sigma\) of \(X\). b) Find the probability that \(X\) differs from its mean by more than one standard deviation.

A multiple-choice test has \(8\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A student guesses independently and randomly on every question. a) Find the probability that the student answers exactly \(2\) questions correctly. b) Find the probability that the student answers at most \(1\) question correctly. c) Which event is more likely: exactly \(2\) correct answers or more than \(2\) correct answers? Support your conclusion by comparing the probabilities.

At a glassworks, each figurine independently has a \(10\%\) chance of being defective. A random sample of \(12\) figurines is selected for quality control. a) Find the probability that none of the selected figurines are defective. b) Find the probability that at least \(2\) figurines are defective. c) Which event is more likely: exactly \(1\) figurine is defective or more than \(2\) figurines are defective?

A basketball player independently makes each free throw with probability \(0.70\). During practice, the player attempts \(6\) free throws. Find the probability of each event. a) The player makes exactly \(4\) free throws. b) The player makes no free throws. c) The player makes at most \(1\) free throw. d) The player makes at least \(5\) free throws.

A multiple-choice test has \(6\) questions. Each question has two answer choices, exactly one of which is correct. A person guesses randomly and independently on every question. Find the probability that (1) exactly \(3\) answers are correct, (2) exactly \(5\) answers are correct, (3) more answers are correct than incorrect.

A basketball player makes each free throw independently with probability \(p=0.70\). The player attempts \(4\) free throws. a) Find the probability that the player (1) makes exactly \(2\) free throws and (2) makes at least \(3\) free throws. b) After training, the player's success probability increases to \(p=0.75\). Recalculate the probabilities from part a) and compare them with the original values.

A fair six-sided die is rolled independently \(10\) times. Rolling a \(6\) is considered a success. Find the probability that (1) exactly \(2\) rolls are \(6\)s, (2) at most \(1\) roll is a \(6\), (3) at least \(3\) rolls are \(6\)s.

A gardener plants \(12\) flower seeds of one variety. Each seed germinates with probability \(80\%\), independently of the others. Find the probability that (1) exactly \(10\) seeds germinate, (2) more than \(10\) seeds germinate, (3) at most \(2\) seeds do not germinate.

A spinner has a win probability of \(0.25\) on each spin. It is spun independently \(10\) times. Find the probability of winning exactly \(3\) times or exactly \(4\) times.

Two fair six-sided dice are rolled together, and this experiment is repeated independently \(8\) times. Find the probability that a sum of exactly \(11\) occurs fewer than \(2\) times.

A manufacturer claims that only \(1\%\) of its USB drives are defective. A distributor tests \(100\) drives and observes \(3\) defective drives. Model the tested drives as independently defective with probability \(0.01\). Find the probability, under the claim, of observing at least \(3\) defective drives. The claim will be rejected if this probability is at most \(5\%\). Evaluate the result using this rule.

A survey reports that \(40\%\) of households in a city regularly use a streaming service. A market-research study randomly selects \(15\) households. Assume the household responses are independent. Find the probability that a) exactly \(6\) households use a streaming service, b) at most \(2\) households use a streaming service, c) at least \(13\) households use a streaming service.

An archer hits the center ring of a target with probability \(0.4\) on each shot. She takes \(10\) independent shots. Let \(X\) be the number of center-ring hits. Find each probability. a) Exactly \(4\) hits b) At least \(2\) hits c) Between \(3\) and \(5\) hits, inclusive

In a manufacturing process, \(5\%\) of electronic components are defective. A random sample of \(15\) components is selected, and defect outcomes are treated as independent. Let \(X\) be the number of defective components. Find each probability. a) Exactly \(1\) component is defective. b) More than \(2\) components are defective. c) At least \(1\) but at most \(3\) components are defective.

A player wins each spin of a game wheel with probability \(0.2\). The wheel is spun \(10\) times independently. Let \(X\) be the number of wins. Find the probability of winning at least once but no more than three times.

A basketball player makes a free throw with probability \(0.6\). He takes \(15\) independent free throws. Let \(X\) be the number made. Find the probability that he makes between \(8\) and \(10\) free throws, inclusive.

In a large city, \(60\%\) of commuters drive to work. A random sample of \(20\) commuters is selected, and responses are treated as independent. Let \(X\) be the number who drive. a) What value of \(X\) is most likely? b) Find the probability that at most \(8\) or at least \(16\) commuters in the sample drive.

A gardener knows that \(20\%\) of a certain type of flower seed does not germinate. She plants \(20\) seeds, and outcomes are treated as independent. Let \(X\) be the number that do not germinate. a) What value of \(X\) is most likely? b) Show that the probability that at most \(1\) seed or at least \(8\) seeds do not germinate is about \(10\%\).

A spinner has \(3\) equal sections labeled \(1\), \(2\), and \(3\). A player suspects that the spinner is biased so that \(3\) occurs less often than it should. In \(30\) independent spins, the spinner lands on \(3\) only \(4\) times. 1. Assuming the spinner is fair, find the probability that \(3\) occurs at most \(4\) times. 2. A result is considered significantly rare when the probability of that result or a more extreme result is less than \(5\%\). Under this criterion, does the observation support the player's suspicion? Explain.

A spinner has four equal sections: one red winning section and three blue losing sections. Thus, \(P(\text{red})=0.25\). Assume successive spins are independent. Determine whether each interpretation is correct. (1) If the spinner is spun \(4\) times, it is certain to land on red exactly once. (2) If the spinner does not land on red in the first \(10\) spins, the probability of red on the \(11\)th spin increases to make up for the earlier results. (3) Over a very large number of spins, such as \(10{,}000\), the relative frequency of red will likely be close to \(0.25\). (4) In \(40\) spins, getting exactly \(10\) red results is more likely than getting exactly \(11\) red results.

An electronics manufacturer has found that \(5\%\) of its microchips are defective. Assume that chip defects are independent and that every chip has the same probability of being defective. Determine whether each statement is correct. (1) In a sample of \(100\) chips, the expected number of defective chips is \(5\). (2) The probability that none of the chips in a package of \(20\) is defective is \((0.95)^{20}\). (3) If \(3\) of \(20\) sampled chips are defective, this proves that the stated \(5\%\) defect rate must be wrong because the sample defect rate is \(15\%\). (4) A person stakes \(\$1\) on a randomly selected chip being defective. For a fair wager, the other person should stake \(\$19\) on the chip not being defective.

A spinner has \(10\) equal sections labeled \(1\) through \(10\) and is spun twice. The random variable \(X\) is the number of spins that land on a multiple of \(3\). a) Create a probability distribution table for \(X\). b) Find the probability that the first or second result, or both, is divisible by \(3\). c) Compare the result from part b) with the probability that exactly one result is divisible by \(3\). Which event is more likely?

A basketball player independently takes \(5\) free throws and makes each free throw with probability \(p=0.7\). Find the probability that the player: 1. makes exactly \(3\) free throws; 2. makes at least \(4\) free throws.

A spinner has \(5\) equal sections, and \(1\) section is blue. The spinner is spun \(6\) times independently. Let \(X\) be the number of times it lands on blue. Find each probability. (1) Exactly \(2\) times (2) At most \(1\) time (3) At least \(1\) time

In a microchip manufacturing process, \(5\%\) of chips are defective. A random sample of \(10\) chips is tested, and outcomes are treated as independent. Let \(X\) be the number of defective chips. Find each probability. (1) Exactly \(1\) chip is defective. (2) No chips are defective. (3) More than \(2\) chips are defective.

A multiple-choice test has \(8\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A student guesses randomly on every question. Let \(X\) be the number of correct answers. a) Find the probability of exactly \(3\) correct answers. b) Find the probability of more than \(6\) correct answers.

A person claims to be able to predict whether a face-down playing card is red or black. In a test, \(12\) cards are shown one at a time. After each guess, the card is replaced and the deck is shuffled. Assume the person has no special ability and guesses randomly. Let \(X\) be the number of correct color guesses. a) Find the probability of exactly \(9\) correct guesses. b) Find the probability of at least \(10\) correct guesses.

Ten equally skilled runners compete in a race with no ties. A spectator randomly predicts the runners who will finish first and second, in order. a) Find the probability \(p\) that one random prediction is completely correct. b) Sixty people make independent random predictions. Find the probability that exactly one prediction is correct.

A factory produces electronic components with a defect rate of \(5\%\), and component statuses are independent. A quality-control sample of \(20\) components is selected. Let \(X\) be the number of defective components in the sample. a) Find the probability that exactly \(2\) components are defective. b) Find the probability that at most one component is defective. c) Calculate \(\binom{20}{2}\) and explain its meaning in this context.

A multiple-choice test has \(10\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A student guesses randomly on every question. Let \(X\) be the number of correct answers. a) Find the probability of exactly \(3\) correct answers. b) How many correct/incorrect patterns contain exactly \(2\) correct answers? Use a binomial coefficient. c) Find the probability of at least \(1\) correct answer.

A Bernoulli process has \(n=10\) independent trials with success probability \(p=0.5\). The random variable \(X\) is the number of successes. a) Determine \(P(X=2)\) and \(P(X=8)\). b) Use the binomial probability formula and a property of binomial coefficients to explain why the two probabilities are equal.

In a game show, a player chooses \(4\) numbers from \(18\). The player wins the grand prize only if all \(4\) chosen numbers match the \(4\) numbers drawn. Find the probability that a player wins the grand prize at least once in \(1000\) independent rounds, using one entry per round. Give the result as a percent rounded to two decimal places.

A basketball player makes a free throw with probability \(0.75\). During practice, he takes \(6\) independent free throws. Let \(X\) be the number made. a) Find the probability that he makes exactly \(4\) free throws. b) Find the probability that he makes at least \(5\) free throws.

A spinner has \(20\) equal sections: \(10\) blue, \(6\) red, and \(4\) yellow. The spinner is spun \(15\) times independently. Find each probability. a) Exactly \(5\) blue results b) No yellow results c) More than \(2\) red results

A basketball player makes a free throw with probability \(p=0.75\). During practice, the player takes \(4\) free throws. The random variable \(X\) is the number of made free throws. a) Create a probability distribution table for \(X\). b) Find the probability that the player makes at least \(3\) free throws.

Historically, \(10\%\) of the microchips produced by a factory are defective. A sample of \(5\) chips is selected from current production. The random variable \(Y\) is the number of defective chips in the sample. a) Determine the probability distribution of \(Y\). b) Calculate the expected value \(E(Y)\).

A spinner has four equal sections: one green section labeled “win” and three red sections labeled “loss.” The spinner is spun \(4\) times. a) Determine the probability distribution of the random variable \(X\), the number of wins. b) Determine the probability distribution of the random variable \(Y\), the number of losses. c) Compare the distributions. What relationship holds between \(P(X=k)\) and \(P(Y=4-k)\)?

A seed test finds that each seed of a certain type germinates independently with probability \(0.90\). A gardener plants \(10\) of these seeds. Let \(X\) be the number of seeds that germinate and \(Y\) be the number that do not germinate. a) Find the probability that at least \(9\) seeds germinate. b) Find the probability that at most \(1\) seed does not germinate. c) Explain without calculation why the answers to parts a) and b) must be equal.

At a garden center, \(10\%\) of tulip bulbs do not sprout. Among the bulbs that sprout, \(60\%\) bloom red, \(30\%\) bloom yellow, and \(10\%\) bloom white. One bulb is selected at random. a) Explain why recording the four outcomes—does not sprout, red, yellow, or white—is not a Bernoulli trial. b) Describe two different ways to define success so that the experiment becomes a Bernoulli trial. Find the success probability for each definition.

A fair eight-sided die labeled \(1\) through \(8\) is rolled independently \(10\) times. Let \(X\) be the number of times an \(8\) is rolled. Find the probability of each event. (1) At least \(1\) eight is rolled. (2) More than \(2\) eights are rolled. (3) At most \(9\) eights are rolled. (4) Every roll is an eight.

A spinner has \(5\) equal sections, exactly \(1\) of which is red. The spinner is spun independently \(12\) times. Let \(Y\) be the number of times it lands on red. Find the probability of each event. (1) At least \(1\) red result. (2) More than \(3\) red results. (3) At most \(10\) red results. (4) No red results.

A machine produces parts, and \(20\%\) are defective. A sample of \(6\) parts is selected independently from ongoing production. The random variable \(X\) is the number of defective parts in the sample. a) Give the probability distribution of \(X\) in a table. b) Find each probability: (1) At most \(2\) parts are defective. (2) At least \(1\) part is defective.

A multiple-choice test has \(10\) questions. Each question has \(2\) answer choices, exactly one of which is correct. A student guesses randomly on every question. Let \(X\) be the number of correct answers. a) Find the probability of exactly \(5\) correct answers. b) Find the probability of at least \(8\) correct answers. c) Is exactly \(5\) correct answers or more than \(5\) correct answers more likely? Justify your decision without calculating every point probability.

A faulty computer system generates a random sequence of \(8\) independent bits, where each bit is \(1\) with probability \(0.5\). Let \(X\) be the number of ones in the bit sequence. a) First consider a shorter sequence of \(3\) independent bits. Create a table showing the probability distribution of \(X\). b) For the original \(8\)-bit sequence, find the probability that the number of zeros is strictly greater than the number of ones.

Consider \(5\) independent Bernoulli trials with success probability \(p=\frac{1}{3}\). a) Determine the probability distribution of the random variable \(X\), the number of successes. Give the probabilities as fractions. b) Explain the relationship between the distribution of \(X\) and the binomial theorem using \(\left(\frac{1}{3}+\frac{2}{3}\right)^5\).

A multiple-choice test has \(4\) questions, each with \(5\) answer choices and exactly one correct answer. A student guesses independently on every question. The random variable \(X\) is the number of correct answers. a) Create a probability distribution table for \(X\). b) Find the probability of at least two correct answers. c) Which binomial coefficient counts the answer patterns with exactly two correct answers? Evaluate it.

A manufacturer finds that each Bluetooth headphone independently develops a battery defect during the warranty period with probability \(0.15\). A sample of \(12\) headphones is selected. Find the probability that a) exactly \(3\) headphones have a battery defect, b) at most \(2\) headphones have a battery defect, c) more than \(4\) headphones have a battery defect.

In a large city, \(40\%\) of commuters regularly ride a bicycle to work. A sample of \(15\) commuters is surveyed. Assume the commuter responses are independent. Find the probability that a) exactly \(6\) regularly ride a bicycle to work, b) fewer than \(5\) regularly ride a bicycle to work, c) at least \(8\) regularly ride a bicycle to work.

A target shooter independently takes \(5\) shots in each series and hits the target with probability \(0.80\) on each shot. A five-shot series is called highly successful if the shooter gets at least \(4\) hits. The shooter completes \(12\) independent five-shot series. Find the probability that: 1. one series is highly successful; 2. exactly \(8\) of the \(12\) series are highly successful; 3. none of the \(12\) series is highly successful.

In a large city, each teenager independently owns a tablet with probability \(0.80\). A sample of \(50\) teenagers is selected. Let \(X\) be the number who own a tablet, and let \(Y=50-X\) be the number who do not own a tablet. Use \(Y\sim\operatorname{Bin}(50,0.20)\) to find the probability of each event. (1) Exactly \(42\) teenagers own a tablet. (2) At least \(38\) teenagers own a tablet. (3) At most \(44\) teenagers own a tablet.

An online retailer finds that \(75\%\) of its customers place orders by smartphone. A random sample of \(100\) completed orders is analyzed, and order types are treated as independent. Let \(X\) be the number placed by smartphone. Find each probability. (1) Fewer than \(70\) smartphone orders (2) Between \(72\) and \(80\) smartphone orders, inclusive (3) More than \(80\) smartphone orders

A manufacturer knows that \(5\%\) of its electronic components are defective. A random sample of \(200\) components is selected, and defect outcomes are treated as independent. Let \(X\) be the number of defective components. Find each probability. (1) At most \(10\) components are defective. (2) At least \(8\) and at most \(12\) components are defective. (3) More than \(15\) components are defective.

A multiple-choice test has \(50\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A student guesses randomly on every question. Let \(X\) be the number of correct answers. Find each probability. (1) At least \(15\) correct answers (2) Fewer than \(10\) correct answers (3) Between \(10\) and \(15\) correct answers, inclusive

A multiple-choice test has \(60\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A student guesses randomly on every question. Let \(X\) be the number of correct answers. Find each probability. a) More than \(12\) correct answers b) At least \(10\) and at most \(20\) correct answers c) Fewer than \(40\) incorrect answers

In a region, \(12\%\) of households no longer have a home landline. A random sample of \(100\) households is selected, and household statuses are treated as independent. Let \(X\) be the number without a landline. a) Find each probability. (1) At most \(10\) households have no landline. (2) More than \(15\) households have no landline. (3) Between \(8\) and \(14\) households, inclusive, have no landline. b) Find the probability that at least \(90\) households still have a landline.

A gardener plants \(60\) seeds of a flower variety. Each seed germinates with probability \(0.75\), independently of the others. Let \(X\) be the number that germinate. Find each probability. 1. At most \(40\) seeds germinate. 2. More than \(50\) seeds germinate. 3. At least \(42\) and at most \(48\) seeds germinate.

In a microchip manufacturing process, \(5\%\) of chips are defective. A random sample of \(100\) chips is tested, and defect outcomes are treated as independent. Let \(X\) be the number of defective chips. Find each probability. 1. Exactly \(5\) chips are defective. 2. At least \(3\) but fewer than \(8\) chips are defective. 3. More than \(10\) chips are defective.

An urn contains one red ball and two white balls. A ball is drawn repeatedly with replacement, and drawing red counts as a success. Consider these events: \(E_1\): Exactly \(1\) success in \(3\) draws. \(E_2\): Exactly \(2\) successes in \(3\) draws. \(E_3\): Exactly \(2\) successes in \(6\) draws. a) Calculate \(P(E_1)\) and \(P(E_2)\), and show that \(P(E_1)=2P(E_2)\). b) Determine whether \(P(E_1)\) is also exactly twice \(P(E_3)\).

Let \(X\sim\operatorname{Bin}(50,0.25)\). Find each probability. a) \(P(X\le 15)\) b) \(P(X\ge 10)\) c) \(P(8\le X\le 14)\)

Let \(X\sim\operatorname{Bin}(100,0.8)\). Find each probability. a) \(P(X\le 75)\) b) \(P(X>85)\) c) \(P(78<X<88)\)

A basketball player independently makes each free throw with probability \(75\%\). During practice, the player attempts \(20\) free throws. Find the probability that (1) every free throw is made, (2) exactly \(15\) free throws are made, (3) more than \(15\) free throws are made.

A factory produces electronic components. Each component is independently defective with probability \(0.05\). A sample of \(80\) components is selected for quality testing. Find the probability that the sample contains (1) no defective components, (2) exactly \(4\) defective components, (3) fewer than \(4\) defective components.

A basketball player attempts \(25\) independent shots and makes each shot with probability \(45\%\). Let \(X\) be the number of made shots. For each event, state its complement in words and find the probability of the complement. (1) \(E_1\): More than \(10\) made shots. (2) \(E_2\): At most \(14\) made shots.

In a survey of \(50\) people, each person independently supports a statement with probability \(0.2\). Let \(X\) be the number who support it. For each event, describe the complement and find the complement's probability. (1) \(E_1\): At least \(12\) people support the statement. (2) \(E_2\): More than \(5\) but at most \(12\) people support the statement.

A new neighborhood has \(150\) households that use the same internet provider. At a randomly selected time during the evening, each household independently uses the service at full bandwidth with probability \(0.20\). Let \(X\) be the number using full bandwidth at that time. What is the minimum number of households the network must be able to serve at full bandwidth simultaneously so that demand is met with probability at least \(95\%\)?

In a quality-control process, each electronic component is assumed to be independently defective with probability \(0.06\). A sample of \(150\) components is selected. Let \(X\) be the number of defective components. Find the probability that a) exactly \(9\) components are defective, b) at most \(5\) components are defective, c) at least \(12\) components are defective.

In a region, \(25\%\) of households use solar energy to generate electricity. A random sample of \(80\) households is selected, and household statuses are treated as independent. Let \(X\) be the number that use solar energy. Find each probability. a) More than \(20\) households use solar energy. b) Fewer than \(15\) households use solar energy. c) At least \(18\) but at most \(22\) households use solar energy.

A hotel has \(80\) rooms. Historical data show that \(10\%\) of guests with reservations do not arrive. For one night, the hotel accepts \(85\) reservations. Let \(X\) be the number of guests who arrive, and treat guest arrivals as independent. Find each probability. 1. At most \(80\) guests arrive, so everyone receives a room. 2. More than \(80\) guests arrive. 3. Exactly \(80\) guests arrive. 4. At least \(70\) but at most \(75\) guests arrive.

A computer-chip manufacturer reports that \(4\%\) of its chips are defective. A random sample of \(200\) chips is tested, and defect outcomes are treated as independent. Let \(X\) be the number of defective chips. Find each probability. 1. At most \(5\) chips are defective. 2. More than \(10\) chips are defective. 3. At least \(6\) but at most \(12\) chips are defective. 4. The number of defective chips equals the mean of \(X\).

A gardener plants \(60\) seeds of a flower variety. Each seed germinates with probability \(0.8\), independently of the others. Let \(X\) be the number that germinate. Find each probability. (1) Exactly \(50\) seeds germinate. (2) Fewer than \(45\) seeds germinate. (3) At least \(52\) seeds germinate. (4) Between \(46\) and \(54\) seeds germinate, inclusive.

In a large population, \(12\%\) of people are left-handed. A random sample of \(150\) people is selected, and handedness outcomes are treated as independent. Let \(X\) be the number who are left-handed. Find each probability. (1) Exactly \(18\) people are left-handed. (2) At most \(15\) people are left-handed. (3) More than \(20\) people are left-handed. (4) At least \(10\) and at most \(25\) people are left-handed.

A gardener plants \(80\) flower bulbs of one variety. Each bulb independently fails to sprout with probability \(0.15\). The gardener needs at least \(65\) flowers for a garden design. Find the probability that the \(80\) planted bulbs produce at least \(65\) flowers.

A collectible card company reports that one out of every seven sealed packs contains a limited-edition card. A random sample of \(20\) packs is selected from a very large production run. a) Explain why this process can be modeled as a binomial setting. b) Identify \(n\) and \(p\) when \(X\) is the number of packs containing a limited-edition card. c) Find the probability that exactly \(3\) of the \(20\) packs contain a limited-edition card. Give the expression and round the result to four decimal places.

A spinner has four equal sections, exactly one of which is red. The spinner is spun \(40\) times. The random variable \(X\) is the number of times the spinner lands on red, so \(p=0.25\). a) Calculate the expected value \(\mu\). b) A binomial distribution has its greatest probability at \(X=\mu\) when \(\mu\) is an integer. Calculate \(P(X=10)\). c) Calculate \(P(X=9)\) and \(P(X=11)\), and compare them with \(P(X=10)\).

A treatment is effective for a patient with probability \(p=0.7\). In a study, the treatment is given to \(15\) patients, and the outcomes are assumed to be independent. a) Find the expected value \(\mu\) and the failure probability \(q=1-p\). b) When \(\mu\) is not an integer, a mode \(k\) lies in the interval \([\mu-q, \mu+p]\). Find this interval and identify the integer \(k\) in it. c) Calculate \(P(X=k)\), \(P(X=k-1)\), and \(P(X=k+1)\) to verify that \(k\) is the most likely number of patients for whom the treatment is effective.

At a large university, \(2.5\%\) of students have season tickets for the local basketball team. A lecture includes \(80\) randomly selected students, and ticket-holder statuses are treated as independent. Let \(X\) be the number who have season tickets. a) Find the probability that exactly \(2\) students have season tickets. b) Find the probability that at most \(3\) students have season tickets. c) Find the probability that at least \(1\) student has a season ticket.

A manufacturer reports that \(0.8\%\) of its microchips are defective. A random sample of \(300\) chips is tested, and defect outcomes are treated as independent. Let \(X\) be the number of defective chips. a) Find \(P(X=0)\), \(P(X=1)\), and \(P(X=2)\). b) Find the probability that more than \(3\) chips are defective. c) Suppose a sample of \(300\) chips is tested every day for \(365\) days. On how many days should the inspector expect to find no defective chips?

At a carnival prize booth, each ticket wins the grand prize with probability \(\frac{1}{200}\). A visitor buys \(200\) tickets. Assume the ticket outcomes are independent and have the same win probability. a) Find the probability that the visitor wins no grand prizes. Compare it with \(\frac1e\approx0.3679\). b) Find the probability that the visitor wins exactly one grand prize. c) Find the probability that the visitor wins at least two grand prizes.

At a coworking space, \(15\) members are present. Each member uses a soundproof phone booth for an average of \(15\) minutes during an hour. Assume the members' booth-use times are independent and uniformly distributed throughout the hour. At a randomly selected time, let \(X\) be the number of booths in use. a) Find the probability that at most \(4\) booths are needed. b) Find the probability that more than \(6\) booths are needed. c) What is the minimum number of booths needed so that all members can be accommodated with probability at least \(95\%\)?

In a warehouse, \(20\) employees work at packing stations. Each employee uses a powered pallet jack for an average of \(6\) minutes per hour. Assume the employees' use times are independent and uniformly distributed throughout the hour. At a randomly selected time, let \(X\) be the number of pallet jacks needed. a) Find the probability that exactly \(2\) pallet jacks are needed. b) Find the probability that more than \(4\) pallet jacks are needed. c) What is the minimum number of pallet jacks that must be available so that demand is met with probability at least \(99\%\)?

A highway rest area has \(6\) fast-charging stations for electric vehicles. During a two-hour period, \(24\) vehicles arrive to charge. Assume that, at a randomly selected time during the period, each vehicle independently has probability \(\frac{20}{120}=\frac{1}{6}\) of needing a charging station. Find the probability that more than half of the charging stations are needed at that time.

An office has \(50\) employees who share licenses for specialized software. Each employee uses the software for an average of \(15\) minutes per hour. Assume access times are independent and uniformly distributed throughout the hour. At a randomly selected time, let \(X\) be the number of employees using the software. What is the minimum number of licenses needed so that the probability that at least one employee cannot access the software is at most \(2\%\)?

A fitness studio offers a circuit class for \(40\) participants. One station has \(10\) identical rowing machines. Each participant uses a rowing machine for a total of \(10\) minutes during the \(60\)-minute class. Assume the participants' use times are independent and uniformly distributed throughout the class. At a randomly selected time, let \(X\) be the number who want to use a rowing machine. a) Find the probability that more than \(10\) participants want to use a rowing machine at the same time. b) What is the minimum total number of rowing machines needed so that the probability that someone must wait is at most \(5\%\)?

A medical office has a waiting room with \(10\) seats. The office is open for \(300\) minutes each morning and schedules \(80\) patients. Assume that each patient spends \(15\) minutes in the waiting room and that the waiting periods are independently and uniformly distributed over the morning. Thus, the probability that a given patient is in the waiting room at a randomly selected time is \(\frac{15}{300}=0.05\). Find the probability that more patients are in the waiting room than there are seats. Then determine whether \(10\) seats are sufficient under a requirement that overcrowding occur no more than \(1\%\) of the time.

A component passes a rigorous quality inspection with probability \(p=0.3\). In a batch of \(n=600\) independently tested components, let \(X\) be the number that pass. The mean is \(\mu=180\). Find the probability that \(X\) is outside the interval \([165,195]\).

A binomial random variable \(X\) has parameters \(n=150\) and \(p=0.4\). 1. Find the mean \(\mu\) and standard deviation \(\sigma\). 2. Find the probability that \(X\) is within \(5\) of its mean.

A binomial random variable \(X\) has parameters \(n=200\) and \(p=0.4\). First find the mean \(\mu\). Then state how far each endpoint of the interval \([72,90]\) is from the mean and find \(P(72\le X\le 90)\).

At a distribution center, automated scanners cannot read the barcode on \(2\%\) of packages. A random sample of \(400\) packages is inspected, and barcode outcomes are treated as independent. Let \(X\) be the number with unreadable barcodes. Find each probability. a) At most \(5\) packages have unreadable barcodes. b) More than \(10\) packages have unreadable barcodes. c) Between \(6\) and \(12\) packages, inclusive, have unreadable barcodes.

A pharmaceutical company reports that a certain side effect occurs in \(40\%\) of patients who take a medication. A study includes \(100\) patients, and side-effect outcomes are treated as independent. Let \(X\) be the number who experience the side effect. Find the probability that the number of patients with the side effect is: a) less than \(35\); b) at least \(45\); c) within \(5\) of the mean.

A factory produces components with a \(4\%\) defect rate. A random sample of \(500\) components is selected, and defect outcomes are treated as independent. Let \(X\) be the number of defective components in the sample. a) Find the probability that at most \(15\) components are defective. b) Find the probability that at least \(25\) components are defective. c) Find the probability that between \(18\) and \(22\) components, inclusive, are defective.

At a light-bulb factory, \(4\%\) of bulbs have a minor cosmetic defect. Assume defects occur independently. a) Find the probability that at least one bulb in a random box of \(25\) bulbs has a cosmetic defect. b) What is the minimum number of bulbs that must be inspected so that the probability of finding at least one bulb with a cosmetic defect is greater than \(90\%\)?

A computer company receives USB drives in packages of \(50\). Each drive is independently defective with probability \(0.10\). A customer needs \(42\) working drives for a training session and opens a new package. Find the probability that the package contains enough working drives.

At a regional blood donation center, \(15\%\) of donors have type O-negative blood. Assume the blood types of \(200\) expected donors are independent. Let \(X\) be the number with type O-negative blood. a) Find the probability that at most \(25\) donors have type O-negative blood. b) Find the probability that more than \(35\) donors have type O-negative blood. c) Find the probability that more than \(25\) but fewer than \(40\) donors have type O-negative blood.

An online retailer emails a promotional offer to \(120\) customers. Each customer independently places an order with probability \(0.12\). Let \(X\) be the number of orders placed. a) Find the probability that fewer than \(10\) orders are placed. b) Find the probability that at least \(15\) customers place an order. c) Find the probability that between \(12\) and \(20\) orders, inclusive, are placed.

Historically, \(4\%\) of the microchips produced by a factory are defective. Ten chips are selected from a very large shipment. Because the population is very large, defect outcomes are modeled as approximately independent. a) Explain why a binomial model is reasonable even though the chips are selected without replacement. b) Find the probability that exactly one selected chip is defective. c) Find the probability that at most one selected chip is defective.

Let \(X\) be a binomial random variable with parameters \(n\) and \(p\). a) Use sigma notation to write the sum of \(P(X=k)\) for \(k=0,1,\ldots,n\). b) Use the binomial theorem \((a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^kb^{n-k}\) to explain why the probabilities in any binomial distribution add to \(1\) for every \(p\in[0, 1]\).

Consider the expression \(T=\sum_{k=0}^{12}\binom{12}{k}(0.25)^k(0.75)^{12-k}\). a) Interpret this expression in the context of Bernoulli trials. Identify \(n\) and \(p\), and explain what \(T\) represents. b) Find the exact value of \(T\) without evaluating the terms individually. Use the binomial theorem. c) How would the value change if the sum ran from \(k=1\) to \(k=12\)? Give an expression for the new value.

At a nursery, each seed of a certain flower variety independently germinates with probability \(p=0.80\). A total of \(20\) seeds are planted, and \(X\) is the number that germinate. Find \(P(X\le 14)\) by introducing a complementary binomial random variable with success probability at most \(0.5\).

An online clothing retailer finds that each ordered item is independently returned because of fit with probability \(0.30\). A sample of \(50\) orders is examined. Find the probability that 1. exactly \(15\) items are returned, 2. fewer than \(10\) items are returned, 3. more than \(20\) items are returned.

A manufacturer finds that \(5\%\) of its microchips are defective. A random sample of \(200\) chips is inspected, and defect outcomes are treated as independent. Let \(X\) be the number of defective chips. Find each probability. 1. At most \(12\) chips are defective. 2. Between \(5\) and \(15\) chips, inclusive, are defective. 3. The number of defective chips is within \(2\) of the mean.

A bag contains \(25\) distinct numbered balls. A ball is drawn at random and replaced, and this process is repeated for a total of \(25\) draws. Focus on one specified ball, such as the ball numbered \(7\). a) Find the probability that this ball is drawn zero times, exactly once, and exactly twice. b) Find the probability that the ball is drawn more than twice. c) Find the expected number of times the ball is drawn.

The spinner shown has \(5\) equal sections: \(2\) blue and \(3\) yellow. The spinner is spun independently \(5\) times. Let \(X\) be the number of times it lands on blue. a) Find \(P(X\le 1)\). b) Find \(P(X\ge 3)\). Give each result as an exact decimal and as a percent.

The spinner shown has \(5\) equal sections. It is spun independently \(6\) times, and \(X\) is the number of times it lands on green. a) Explain why \(X\) has a binomial distribution, and state \(n\) and \(p\). b) Find the probability that the spinner lands on green exactly \(2\) times. c) Find the probability that the spinner lands on green at most \(1\) time.

The spinner shown has \(6\) equal sections: \(3\) green, \(2\) yellow, and \(1\) red. The spinner is spun independently \(8\) times. a) Find the probability that it lands on green at most \(3\) times. b) Find the probability that it lands on red at least \(2\) times. Give each answer as a simplified fraction and as a percent rounded to two decimal places.

The spinner shown has \(10\) equal sections. It is spun \(30\) times. Find each probability. Round each answer to the nearest hundredth of a percent. a) The spinner lands on red at most \(8\) times. b) The spinner lands on blue at least \(12\) times and at most \(18\) times.

A spinner has \(10\) equal sections: \(5\) blue, \(3\) yellow, and \(2\) red. The spinner is spun independently \(12\) times. Find the probability that a) it lands on blue at most \(4\) times, b) it lands on red at least \(2\) times.

The spinner shown has \(8\) equal sections and is spun independently \(10\) times. Let \(X\) be the number of times it lands on red. a) Explain why \(X\) has a binomial distribution, and state \(n\) and \(p\). b) Find \(P(X\le 2)\). c) The spinner is now spun independently \(4\) times. Find the probability that blue occurs exactly \(2\) times and that the two blue results are consecutive.

The spinner shown has \(8\) equal sections and is spun independently \(3\) times. Find which expressions correctly calculate the probability that the spinner lands on yellow exactly twice. Explain your choices. (1) \(\left(\frac38\right)^2\frac58\) (2) \(3\cdot\frac38\left(\frac58\right)^2\) (3) \(\frac38\cdot\frac38\cdot\frac58+\frac38\cdot\frac58\cdot\frac38+\frac58\cdot\frac38\cdot\frac38\) (4) \(3\cdot\left(\frac38\right)^2\frac58\)

A spinner has \(6\) equal sections: \(2\) yellow and \(4\) blue, as shown. A researcher wants to test whether the spinner has been biased in favor of blue. The spinner is spun independently \(50\) times, and the number of yellow results is recorded. Assuming the spinner is fair, find the probability that yellow occurs at most \(10\) times. Round the result to three decimal places.

A spinner has \(10\) equal sections: \(3\) purple sections labeled P and \(7\) orange sections labeled O. The spinner is spun independently \(8\) times. Let \(X\) be the number of purple results. For each expression, describe an event in context whose probability it represents. 1) \(\binom{8}{2}(0.30)^2(0.70)^6\) 2) \((0.70)^8\) 3) \(1-(0.70)^8\) 4) \(\sum_{k=0}^{2}\binom{8}{k}(0.30)^k(0.70)^{8-k}\)

The spinner shown has \(8\) equal sections. a) The spinner is spun independently \(50\) times. Find the probability of each event. \(A\): Green occurs exactly \(20\) times. \(B\): Red occurs at most \(10\) times. b) Describe an event in context whose probability is \(\sum_{k=0}^{8}\binom{60}{k}(0.125)^k(0.875)^{60-k}\).

A seed supplier states that its wheat seeds have a \(95\%\) germination rate. A farmer suspects that the actual germination probability is only \(80\%\). To compare the two models, the farmer tests \(20\) seeds and observes that exactly \(17\) germinate. Assume seed outcomes are independent under each model. a) For Model A, \(p=0.95\), and Model B, \(p=0.80\), find the probability that exactly \(17\) of \(20\) seeds germinate. b) Which model makes the observed result more plausible? Explain. c) The diagram shows the first two stages of the sample under both models. Identify the missing probabilities \(p\) and \(q\) for Model B.

A multiple-choice test has \(3\) questions. Each question has \(4\) answer choices, and exactly one choice is correct. A student guesses randomly on every question. The random variable \(X\) is the number of questions answered correctly. a) Find the probability distribution of \(X\) and display it in a table. b) Explain how the probabilities are related to the terms in the binomial expansion of \((0.25+0.75)^3\).

The probability of an event \(B\) is given by \(P(B)=\left(\frac{2}{3}\right)^4+4\cdot\frac{1}{3}\cdot\left(\frac{2}{3}\right)^3+6\cdot\left(\frac{1}{3}\right)^2\cdot\left(\frac{2}{3}\right)^2\). Describe a random experiment that matches this expression, and state event \(B\) in words.

A quality-control inspector checks electronic components, each of which independently has a \(12\%\) chance of being defective. Let \(X\) be the number of defective components in a sample of size \(n\). a) Find the minimum number of components the inspector must check so that the probability of finding at least one defective component is at least \(90\%\). b) Use systematic trial or a table to find the minimum value of \(n\) for which the probability of finding at most one defective component is less than \(10\%\).

A binomial random variable \(X\) has \(n=12\). Find the success probability \(p\) such that \(P(X\le 1)=0.15\). Round your answer to the nearest hundredth.

A binomial random variable \(X\) has \(n=20\). Find the success probability \(p\) if \(P(X\ge 15)=0.05\). Round your answer to the nearest hundredth.

A basketball player has a free-throw success probability of \(p\). Assume the outcomes of the free throws are independent. Find \(p\), rounded to the nearest hundredth, if the probability that the player makes at most \(1\) of \(4\) free throws is \(65\%\).

An athlete wants the probability of making at least \(13\) successes in \(15\) independent attempts to be at least \(75\%\). What is the minimum success probability \(p\) for each attempt? Round to the nearest thousandth.

A quality-control inspector checks samples of \(20\) LED bulbs. Each bulb independently has probability \(p\) of being defective. A sample is accepted only when at most \(1\) bulb is defective. Find the boundary value of \(p\) for which a sample is accepted with probability \(90\%\). Round the boundary value to the nearest thousandth.

The germination probability for each seed of a certain type of flower is \(25\%\), independently of the other seeds. What is the minimum number of seeds a gardener must plant so that the probability that at least \(3\) seeds germinate is at least \(90\%\)?

An archer has the same probability \(p\) of hitting the bull's-eye on each of \(10\) independent shots. The archer wants at least an \(80\%\) chance of hitting the bull's-eye on at least \(8\) shots. Find the threshold value \(p_0\) such that the goal is met when \(p\ge p_0\). Report \(p_0\) to three decimal places. Then, if \(p\) can be specified only in increments of \(0.01\), state the smallest value that meets the goal.

A shipment contains \(20\) electronic components. Each component independently meets quality standards with probability \(p\). The probability that at least \(18\) components meet the standards must be at least \(70\%\). Find the threshold value \(p_0\) such that the requirement is met when \(p\ge p_0\). Report \(p_0\) to three decimal places. Then, if \(p\) can be specified only in increments of \(0.01\), state the smallest value that meets the requirement.

A quality-control inspector examines boxes containing \(50\) glass bottles each. Each bottle independently has probability \(p\) of containing a small air bubble. The probability that a box contains at most \(2\) bottles with air bubbles is exactly \(15\%\). Find \(p\). Round your answer to the nearest thousandth.

Each computer chip produced by a manufacturer independently has a \(4\%\) chance of being defective. A sample of \(20\) chips is selected. a) What is the probability that exactly \(1\) chip in the sample is defective? b) What is the probability that at most \(2\) chips in the sample are defective? c) What is the minimum number of chips that must be checked so that the probability of finding at least \(1\) defective chip is at least \(99\%\)?

A factory produces LED bulbs, and each bulb independently has a \(4\%\) probability of being defective. a) A package contains \(25\) bulbs. What is the probability that at least \(1\) bulb is defective? b) What is the minimum number of bulbs an inspector must test so that the probability of finding at least \(1\) defective bulb is at least \(90\%\)? c) What is the minimum number of bulbs needed so that the probability of finding at least \(2\) defective bulbs is at least \(90\%\)?

During normal operation, \(5\%\) of a factory's components are defective. A random sample of \(20\) components is inspected, and defect outcomes are treated as independent. Let \(X\) be the number of defective components. a) Find the probability that the sample contains at least \(2\) defective components. b) Find the smallest integer \(k\) for which the probability that the sample contains at least \(k\) defective components is less than \(1\%\).

The fill weight \(X\) of coffee packages is normally distributed with mean \(\mu=500\,\text{g}\) and standard deviation \(\sigma=4\,\text{g}\). A package is underweight if it weighs less than \(495\,\text{g}\). 1. Find the proportion of packages classified as underweight. 2. A supermarket receives \(100\) packages. Find the probability that exactly \(10\) are underweight. 3. What is the minimum number of packages in a delivery for the probability of at least one underweight package to exceed \(99\%\)?

A spinner has five equal sections labeled \(1\), \(2\), \(3\), \(4\), and \(5\). It is spun three times, producing results \(x_1\), \(x_2\), and \(x_3\). Define the partial sums \(S_1=x_1\), \(S_2=x_1+x_2\), and \(S_3=x_1+x_2+x_3\). a) Find the probability that none of the three partial sums is divisible by \(5\). b) The three-spin game is played \(10\) times. Find the probability that none of the partial sums is divisible by \(5\) in exactly \(5\) of the games.

Let \(X\sim\operatorname{Bin}\left(60,\frac{1}{3}\right)\). a) Find the probability that \(X\) differs from its mean by at most \(2\). b) Find \(P(|X-20|\ge 4)\). c) Find the smallest nonnegative integer \(k\) for which \(P(|X-20|\le k)>0.5\).

Let \(X\) be a binomial random variable with \(n=14\) and \(p=0.4\). Find every value of \(k\) for which \(P(X=k)\) is greatest. Justify the maximum by analyzing the ratio \(\frac{P(X=k)}{P(X=k-1)}\).

A certain type of flower seed germinates independently with probability \(0.90\). Each packet contains \(12\) seeds. a) Find the probability that at least \(11\) seeds in a randomly selected packet germinate. b) A gardener buys \(5\) packets. Find the probability that at most \(2\) of the packets each have at least \(11\) seeds germinate.

At a distribution center, an automated conveyor system sorts packages with an error rate of \(4\%\). A sample of \(50\) consecutively sorted packages is inspected. 1. Using a binomial model, find the probability that more than \(3\) packages were sorted incorrectly. 2. Using the same model, find the probability that the number of incorrectly sorted packages equals the expected value. 3. Evaluate the independence assumption if sorting errors are often caused by brief mechanical jams that affect several consecutive packages.

A multiple-choice test has \(20\) questions. Each question has \(4\) answer choices, exactly one of which is correct. A test taker guesses independently and randomly on every question. Find the probability of each event. a) Only the first \(5\) answers are correct; all remaining answers are incorrect. b) More than \(15\) answers are incorrect. c) The first \(3\) answers are correct, and exactly \(6\) answers are correct in total. d) At least \(4\) of the first \(5\) answers are correct, and at most \(1\) of the last \(5\) answers is correct.

A medication produces the desired response in \(85\%\) of patients. Assume patient responses are independent. A preliminary study must be large enough that the probability of observing the response in at least \(10\) patients is at least \(90\%\). Find the minimum number \(n\) of patients needed. Show how you used systematic testing or appropriate technology.

A certain flower seed germinates independently with probability \(0.80\). a) Twelve seeds are planted. Find the probabilities of these events: - \(C\): At most \(10\) seeds germinate. - \(D\): Exactly \(2\) seeds do not germinate. b) Explain the meaning of \(1-(0.8)^{12}\) in context. c) Among plants that germinate, \(2\%\) are affected by a particular pest. Assume the pest outcomes are independent. What is the minimum number of seeds that must be planted so that the probability of producing at least \(1\) affected plant is greater than \(95\%\)?

A glass manufacturer finds that each drinking glass independently has an \(8\%\) chance of having small inclusions and being classified as defective. a) Six glasses are inspected in order. Find the probability of each event. \(A\): Only the last \(2\) glasses are defective. \(B\): At most \(2\) glasses are defective. b) A sample of \(50\) glasses is selected. Find the probability that at least \(2\) are defective. c) Three glasses are packed in each box. A box is acceptable if none of its \(3\) glasses is defective. Find the probability that a randomly selected box is acceptable. d) What is the minimum number of three-glass boxes that must be inspected so that the probability of finding at least \(1\) acceptable box is at least \(95\%\)? e) What is the maximum defect probability \(p\) that would make the probability of finding no defective glasses in a box of \(20\) at least \(90\%\)?

A seed company states that a certain variety of vegetable seed has a germination probability of \(75\%\). Assume that the seeds germinate independently. a) A gardener plants \(5\) seeds in a pot. Find the probability that exactly \(4\) seeds germinate. b) Find the probability that at least \(4\) of the \(5\) seeds germinate. c) The gardener plants a row of \(10\) seeds. Find the probability that the first \(3\) seeds germinate and at least \(4\) of the remaining \(7\) seeds germinate. d) What is the minimum number of seeds the gardener must plant so that the probability that at least \(1\) seed germinates is at least \(99\%\)? e) Suppose the germination probability is unknown. What is the minimum value of \(p\) needed so that all \(10\) seeds in a packet germinate with probability at least \(95\%\)?

During basketball practice, a player shoots free throws in sets of \(4\). The player makes each free throw with probability \(0.70\), independently of all other shots. A set is considered successful if the player makes at least \(3\) of the \(4\) free throws. The player completes \(10\) sets. Find the probability that at least \(9\) sets are successful.

A soldering defect occurs in \(5\%\) of electronic components. A random sample is selected for quality inspection, and defect outcomes are treated as independent. Find the minimum sample size \(n\) needed so that the probability of finding at least \(2\) components with soldering defects is at least \(90\%\).

A conservation project estimates that each released juvenile animal of a rare species has a \(15\%\) chance of surviving its first year in the wild. Assume the survival outcomes are independent. What is the minimum number of juvenile animals that must be released so that the probability that at least \(5\) survive the first year is at least \(99\%\)?

A factory produces microchips, and \(8\%\) are defective. A quality-control sample of \(400\) chips is tested sequentially, with outcomes treated as independent. Let \(X\) be the number of defective chips in the sample. a) Find the probability that the twelfth chip tested is the first defective chip. b) Find the probability that \(X\) is within one standard deviation of its mean.

Historically, each microchip produced by a manufacturer independently has an \(8\%\) chance of being defective. A new manufacturing process is intended to reduce the defect rate. To evaluate the process, the manufacturer tests \(250\) chips from the new production line. The process is recognized as a significant improvement if the number of defective chips in the sample is at least \(1.5\) original standard deviations below the original expected number. Find the probability that the improvement is not recognized when the new process has actually reduced the defect rate to \(4\%\).

In a large shipment of electronic components, each component is defective independently with probability \(0.10\). A random sample of \(20\) components is selected. Let \(X\) be the number of defective components. a) Find the mean of \(X\). b) Find the shortest interval of consecutive integer values that contains the mean and has probability greater than \(90\%\).

In a microchip manufacturing process, \(5\%\) of chips are defective, independently of one another. Chips are packaged in groups of \(10\). A package is rejected if it contains at least one defective chip. An inspector examines \(20\) packages. 1. Find the probability that one randomly selected package is rejected. 2. Find the probability that exactly \(5\) of the \(20\) packages are rejected. 3. Find the probability that at most \(1\) of the \(20\) packages is rejected.

Consider a binomial setting with \(n\) trials and success probability \(p=\frac1n\). Let \(X\) be the number of successes. a) Prove that for every \(n>1\), \(P(X=1)=\frac{n}{n-1}P(X=0)\). b) For large \(n\), \(\left(1-\frac1n\right)^n\approx\frac1e\). Rewrite \(P(X=2)\) to show that for large \(n\), \(P(X=2)\approx\frac{1}{2e}\).

For quality control, a glass panel is divided into \(80\) equal sections. The panel contains \(60\) tiny air bubbles that are distributed independently and uniformly across the panel. a) Find the probability that one specified section contains more than \(2\) air bubbles. b) Find the expected number of the \(80\) sections that contain no air bubbles.

A company has \(20\) electric vehicles available through an employee car-sharing program. At a randomly selected time, each participating employee independently needs a vehicle with probability \(0.10\). What is the greatest number of employees who can participate so that the probability that every employee who needs a vehicle can get one is at least \(95\%\)?

An urn contains \(25\%\) red balls and \(75\%\) blue balls. A ball is drawn with replacement \(192\) times. Let \(X\) be the number of red balls drawn and \(Y\) the number of blue balls drawn. 1. Find the standard deviation of each random variable. 2. Use the binomial distribution to find the probability that \(X\) lies within two standard deviations of its mean. 3. Without another probability calculation, explain why the corresponding two-standard-deviation probability for \(Y\) is exactly the same.

A multiple-choice test has \(25\) questions. A well-prepared student answers each question correctly with probability \(0.75\), independently of the other questions. Let \(X\) be the number of correct answers. a) Briefly explain why cumulative probabilities are useful for finding an interval probability such as \(P(15\le X\le 20)\). b) Find \(P(15\le X\le 20)\) by defining \(Y\) as the number of incorrect answers and using \(p=0.25\).