A simulation uses one random digit from \(0\) through \(9\), with all ten digits equally likely. An event should occur with probability \(0.40\). The simulation records the event when the digit is \(0,1,2,\) or \(3\). Is this mapping valid? Explain briefly.

A simulation of \(20\) repeated trials records \(7\) successes. Use the simulation to estimate the probability of success.

Noah wants to estimate how often **both** players make their penalty kicks. In his model, each player has a \(50\%\) chance of scoring, and one kick does not affect the other. He flips two fair coins: heads means a goal and tails means a miss. Does one coin flip or one pair of flips count as one simulation trial? What coin result counts as a successful trial?

A ferry company simulates \(20\) morning crossings. The ferry arrives late in \(7\) trials. Estimate the chance that the ferry arrives late on a morning crossing.

A game has a \(50\%\) chance of awarding a ticket. A class proposes using one spin of the wheel for each game. A spin counts as a ticket win when it lands on a sector marked T. Does the wheel model the ticket chance correctly? Explain.

A finite population has the values \(4,7,9,12,15,18,22,26,31\). A simple random sample of \(5\) distinct values is selected. Describe a simulation to estimate the probability that the sample median exceeds \(15\). State exactly what must be generated and how a trial is classified.

A trial consists of drawing \(3\) tokens without replacement from the urn shown. Describe a simulation that estimates the probability of drawing exactly \(2\) blue tokens. State what must be reset between trials.

Leila estimates a probability by running \(100\) simulated trials at a time. After each batch, she calculates the success percentage so far. She stops as soon as the percentage is above \(60\%\) and reports that result. Why might this give a result that is too high? Describe a fair plan for deciding how many trials to run.

Design a simulation to estimate the probability that at least two of \(8\) randomly selected people share a birth month. Assume all \(12\) months are equally likely and ignore seasonal effects. Specify one trial and the event recorded.

A simulation repeats a trial until the first success or until \(6\) attempts have occurred. In \(2000\) repetitions, the first success occurred on attempts \(1\) through \(6\) with counts \(390,310,252,208,158,130\), and no success occurred in \(552\) repetitions. Estimate the probability that the first success occurs by attempt \(3\), and the probability of no success in \(6\) attempts.

A fair coin is flipped \(12\) times. A simulation records the length of the longest consecutive run of heads in each trial. Describe how one trial should be processed. In \(10{,}000\) trials, the longest run was at least \(4\) heads in \(3118\) trials. Estimate the probability and explain why recording only the total number of heads would be insufficient.

A cereal box contains one of four equally likely puzzle types. Design a simulation to estimate the probability that all four types are collected within \(7\) boxes. State why recording only the number of boxes is not enough.

A simulation program runs \(1000\) trials, but every trial produces exactly the same five pseudorandom numbers. The code calls `set_seed(31415)` at the start of each trial. Diagnose what the seeding instruction is doing. Where should the seed be set if the analyst wants one reproducible simulation run with distinct trials?

Two appointments independently choose start times uniformly between \(1{:}00\) p.m. and \(3{:}00\) p.m. Each appointment lasts \(30\) minutes. Describe a simulation to estimate the probability that the appointments overlap. State the numerical condition for a successful trial.

A planning team considers two plausible models for an event that can occur on each opportunity: one model uses \(p=0.20\) and the other uses \(p=0.30\). Under each model, the team simulates \(1000\) runs of \(5\) independent opportunities and records whether at least one event occurs. The \(p=0.20\) model produced \(671\) successful runs, and the \(p=0.30\) model produced \(834\) successful runs. A colleague says, “The assumed probability changes by only \(0.10\), so the model choice cannot matter much.” Use the simulation output to assess that claim. Also state one design feature that must be held fixed so the comparison measures sensitivity to \(p\) rather than to some other change.

A bookstore uses a fair random digit from \(0\) through \(9\) to simulate one free gift. The intended chances are \(40\%\) for a sticker, \(30\%\) for a pencil, \(20\%\) for a badge, and \(10\%\) for a bookmark. A student assigns digits \(0\)–\(3\) to stickers, \(4\)–\(6\) to pencils, and \(7\)–\(9\) to badges. In \(100\) trials, no bookmarks occur. a) Why is this not a good simulation of the intended gifts? b) Change the digit assignments to simulate all four gifts fairly. Under your corrected model, about how many bookmarks would you expect in \(100\) trials?

Students simulate a raffle in which a player can try up to five times to win. They record **which try produced the first win**, or “none” if all five tries lost. The results of \(50\) simulated games are: <table> <tr><th>First win</th><th>Try 1</th><th>Try 2</th><th>Try 3</th><th>Try 4</th><th>Try 5</th><th>None</th></tr> <tr><td>Games</td><td>\(12\)</td><td>\(8\)</td><td>\(7\)</td><td>\(6\)</td><td>\(4\)</td><td>\(13\)</td></tr> </table> a) Estimate the probability of winning **by try 3** and **by try 5**. b) If students had recorded only “won” or “lost,” which estimate could they still make, and which would they no longer know?

Twelve balls are independently dropped into four equally likely bins. A simulation estimates the probability that no bin receives more than \(4\) balls. State what one trial must generate and what summary it must retain. In \(8000\) trials, \(3126\) met the condition. Estimate the probability.

Four independent runs of the same simulation each use \(100\) trials. The numbers of successes are \(42\), \(51\), \(47\), and \(44\). a) Explain why the four estimated probabilities are not identical even though the simulation procedure is the same. b) Combine all four runs to give one estimated probability of success.

Two valid simulations estimate the same unknown probability. Run A uses \(50\) trials and records \(31\) successes. Run B uses \(2000\) trials and records \(1190\) successes. a) Find the estimated probability from each run. b) If each simulation were repeated many times, which run size would generally produce estimates that vary less from repetition to repetition? Explain without assuming the true probability is known.

To simulate a \(40\%\) chance of rain, Maya generates one random digit from \(0\) through \(9\), with each digit equally likely. Which four digits could she use for rain? Explain why the choice works.

A simulation models a fair coin. In a run of \(20\) flips, heads occurs \(13\) times. In a separate run of \(100\) flips, heads occurs \(48\) times. Find each estimate of the probability of heads. Which estimate is closer to \(\frac{1}{2}\)? Must either run give exactly \(\frac{1}{2}\)?

A prize game awards a sticker \(25\%\) of the time. Liam simulates each play by rolling a fair six-sided die. He calls \(1\) a win and \(2,3,4,5,6\) a loss. Explain the problem with this simulation. Describe a fair chance device that would work instead.

A class uses the balls in the jar to simulate drawing two raffle tickets without replacement. A red ball represents a winning ticket. a) How many draws make one complete trial? b) If the first ball is red, how many red balls remain for the second draw? c) What should students do with the balls before starting the next trial?

Kwame wants to simulate a fair six-sided die using random digits from \(0\) to \(9\). His first rule reports die result \(1\) for digits \(0,6\), result \(2\) for \(1,7\), result \(3\) for \(2,8\), result \(4\) for \(3,9\), result \(5\) for \(4\), and result \(6\) for \(5\). a) Is the simulated die fair? Give the chance of each result. b) Give a fair rule that accepts some digits and tries again after the others. c) What fraction of digits would your rule reject in the long run?

A programmer has one long sequence of \(1000\) simulated coin flips. To create \(991\) “trials” of \(10\) flips, the programmer uses flips \(1\) through \(10\), then \(2\) through \(11\), then \(3\) through \(12\), and so on. The event is “at least \(7\) heads.” Explain why these trials are not independent and describe a corrected design that uses the same \(1000\) flips.

Eight teams are placed uniformly at random into a fixed single-elimination bracket. Teams A and B are the two highest-ranked teams. Describe a simulation to estimate the probability that A and B are placed in opposite halves of the bracket, so their first possible meeting is the championship. What feature of the random placement determines success?

To simulate a voter selecting option A in a mock poll, Yuna uses one random digit from \(0\) through \(9\) and lets \(8\) digits represent the event. The target probability is about \(0.55\). Evaluate the simulation rule and, if needed, propose a better one-digit rule.

A random-number generator returns a real number uniformly from \(0\) up to but not including \(1\). Rafael uses the rule “success if the output is between \(0.00\) and \(0.30\), inclusive.” What probability does the intended interval represent, and why is the word “inclusive” at the upper endpoint irrelevant for this continuous generator but potentially relevant for a generator restricted to hundredths?

A stick of length \(1\) is broken at two independently and uniformly selected points. The three resulting piece lengths form a triangle exactly when the longest piece is shorter than \(\frac12\). Describe a simulation to estimate this probability. Explain why the two break points must be sorted before the three lengths are calculated.

A simulation needs two fair dice to show a sum of at least \(10\). Compare these ways to generate a trial: 1. Roll two dice and start over whenever the sum is less than \(10\). 2. Choose equally likely from only the ordered dice pairs whose sum is at least \(10\). List the possible accepted pairs. Explain why both methods are fair and which method always finishes in one selection.

A token starts at \(0\). On each step it moves \(+1\) with probability \(0.60\) and \(-1\) with probability \(0.40\). A trial stops when the token first reaches \(+3\) or \(-2\). Describe a simulation that estimates the probability of reaching \(+3\) first. Identify the state information that must be updated after every step.

The diagram shows the sampling square and hit region for a Monte Carlo area estimate. Generate points uniformly in the square and count a point as a hit when it lies in the shaded region. In \(50{,}000\) simulated points, \(39{,}241\) were hits. Estimate the shaded area and use it to estimate \(\pi\).

A clinic simulates a screening test. Each trial follows one full path through the probabilities for risk group and test result. a) Describe how to run one trial using two independent, uniformly chosen random numbers from \([0,1)\). State which number intervals give each result. b) The class runs \(200\) trials. Of these, \(36\) end in high-risk positive, \(18\) in low-risk positive, and the rest in negative results. Estimate the share of **positive simulated tests** that came from high-risk cases.

Eight people, including Ava and Ben, sit at random in the eight seats around the circle. Rotations count as the same seating arrangement. Use seat \(S_0\) as a fixed reference seat. a) Describe a simulation that counts each circular arrangement only once. b) How would you decide whether Ava and Ben are next to each other in a trial? c) Why does fixing Ava at \(S_0\) not make any arrangement more likely?

A simulation is intended to estimate the probability that a customer eventually makes a purchase. Each simulated customer is followed until the customer either makes a purchase or leaves. However, trials still active after \(30\) simulated minutes are discarded, and these discarded trials are mostly customers who take a long time to decide. Explain how discarding the unfinished trials can bias the estimated eventual-purchase probability. Give a better simulation rule.

A school carnival is comparing two prize-wheel designs. Design One awards a prize on \(00\)–\(39\) and Design Two on \(00\)–\(49\), using equally likely random two-digit numbers \(00\)–\(99\). The students use the **same** number for both designs in each trial. Their first \(12\) numbers are \(05,47,63,39,41,88,12,44,76,22,50,37\). a) Use these trials to estimate the prize probability for each design. b) How many trials give a prize under Design Two but not Design One? What is the estimated difference between their prize probabilities? c) Based on the assigned number ranges, what is the **theoretical** difference between the prize probabilities? Why may it differ from the simulation estimate?

A school's sound system has a microphone and an amplifier. The table gives the chances that each stops working during one assembly: <table> <tr><th></th><th>Amplifier fails</th><th>Amplifier works</th></tr> <tr><td>Microphone fails</td><td>\(0.08\)</td><td>\(0.06\)</td></tr> <tr><td>Microphone works</td><td>\(0.04\)</td><td>\(0.82\)</td></tr> </table> a) Assign pairs of digits from \(00\) through \(99\) to the four outcomes so that one random pair of digits simulates one assembly. Every pair is equally likely. b) What is the probability that the microphone fails? That the amplifier fails? c) Someone proposes simulating the two failures independently, using just the two probabilities from part b). Would this reproduce the chance that both devices fail in the table? Explain.

A ten-day forecast is modeled conditional on exactly four rainy days, with every four-day subset of the ten days equally likely. a) Describe a direct simulation of one valid ten-day weather schedule. b) Define a success check for the event “at least two rainy days are consecutive.” c) Explain why simulating ten independent rain/no-rain outcomes with probability \(0.4\) does not sample from the stated conditional model.

A game repeatedly rolls a fair six-sided die and adds the results. A trial stops when the running total reaches or exceeds \(12\). The event is “the total equals exactly \(12\) at stopping.” Describe a simulation and explain why overshooting \(12\) is a failure rather than another roll.

Five teams play a round-robin tournament, so every unordered pair of teams plays exactly once. Assume every game is fair, has no tie, and its outcome is independent of every other game. a) Describe one simulation trial for the entire tournament. b) State how to determine whether the trial has an undefeated team. c) Explain why independently generating a final win total for each team would not be a valid simulation.

The communications network shown has five links. In each trial, every link independently works with probability \(0.80\). The network works when at least one complete route connects \(S\) to \(T\). Describe how to simulate one trial without double-counting trials in which more than one route works. In \(25{,}000\) trials, \(22{,}790\) networks worked. Estimate the reliability.

A greenhouse starts in the Normal temperature state. Each hour its state may change, and the chance of each next state depends on the current state. a) Describe a complete trial to simulate whether it is Warm **after exactly six hours**. b) The class runs \(200\) such trials and the greenhouse finishes Warm in \(42\). Estimate the probability of ending Warm. c) Why would using the same fixed chance of Warm on every step give the wrong simulation?

A supposedly fair digit generator produced these counts in \(10{,}000\) trials: <table> <tr><th>digit</th><th>\(0\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th><th>\(5\)</th><th>\(6\)</th><th>\(7\)</th><th>\(8\)</th><th>\(9\)</th></tr> <tr><td>count</td><td>\(1011\)</td><td>\(983\)</td><td>\(1005\)</td><td>\(996\)</td><td>\(1008\)</td><td>\(990\)</td><td>\(1017\)</td><td>\(995\)</td><td>\(1002\)</td><td>\(993\)</td></tr> </table> a) If all digits are equally likely, how many times would you expect each digit in \(10{,}000\) trials? b) Which digits differ most from that expected count, and by how many? c) Are the results close to or far from the fair model? Describe one way to check whether any imbalance keeps appearing.

Inés tries to simulate a point uniformly in the disk shown. Inés generates independent \(U,V\sim\text{Uniform}(0,1)\), sets \(R=U\) and \(\Theta=2\pi V\), and uses polar coordinates \((R,\Theta)\). a) Explain why this method places too many points near the center. b) Derive a corrected formula for \(R\). c) State a complete uniform simulation algorithm for the disk.

A game-show simulation compares “stay” and “switch” in a three-door setup. One door hides a prize. After the contestant chooses a door, the host—who knows the prize location—must open a different door showing no prize. Describe one valid trial that preserves the host’s rule. Explain why choosing the host’s door uniformly from both unchosen doors without checking for the prize can bias the comparison.

A small service desk is simulated minute by minute for \(60\) minutes. At the start of each minute, a customer arrives with probability \(0.35\). If the clerk is busy, the customer joins the end of the line. At the end of each minute, the customer in service finishes with probability \(0.40\). Describe the state information and update order needed for one trial estimating the probability that the line ever reaches \(5\) waiting customers. Explain why simulating only the total numbers of arrivals and completions is not enough.