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Mutually exclusive events

Learning goals

  • Determines whether events are mutually exclusive.
  • Uses the addition rule appropriately.
  • Distinguishes mutually exclusive events from independent events.
  • Finds the probability of either of two disjoint events.

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Mutually exclusive events 1
In a single card draw, let \(A\) be “drawing a heart” and \(B\) be “drawing a club.” Are \(A\) and \(B\) mutually exclusive? Explain using the intersection \(A\cap B\).

Hints

- Can a single playing card be both a heart and a club? - The intersection contains outcomes where both events happen. - What do you write for an event with no possible outcomes?

Solution

1. The experiment records one outcome for a single card draw. 2. The descriptions “drawing a heart” and “drawing a club” cannot both hold for the same outcome. 3. Thus \(A\cap B=\varnothing\), so the events are mutually exclusive.

Answer

Yes. The events are mutually exclusive because \(A\cap B=\varnothing\).
Mutually exclusive events 2
In a single trial, mutually exclusive events \(A\) and \(B\) have \(P(A)=0.2\) and \(P(B)=0.4\). Find \(P(A\cup B)\).

Hints

- Use the addition rule for a union. - Mutual exclusivity makes the intersection probability \(0\). - Add the two given event probabilities after accounting for the overlap.

Solution

1. Mutually exclusive events have \(P(A\cap B)=0\). 2. Therefore \(P(A\cup B)=P(A)+P(B)=0.2+0.4=0.6\).

Answer

\(P(A\cup B)=0.6\).
Mutually exclusive events 3
The circle represents all possible directions. Event \(A\) is the sector on the right between the two solid radii. Event \(B\) is the sector between the two dashed radii. Are \(A\) and \(B\) mutually exclusive? Explain using the diagram.
Figure for problem 547296

Hints

- Identify the sector bounded by the two solid radii. - Compare its location with the sector bounded by the two dashed radii. - Mutually exclusive events cannot share any direction.

Solution

1. The two dashed radii that bound \(B\) lie between the two solid radii that bound \(A\). 2. Therefore every direction in \(B\) is also in \(A\), so \(B\subseteq A\). 3. Thus \(A\cap B=B\), which is not empty. Therefore the events are not mutually exclusive.

Answer

No. Event \(B\) lies inside event \(A\), so \(A\cap B=B\neq\varnothing\).
Mutually exclusive events 4
A two-stage experiment records whether a first draw is red and whether a second draw is blue. Babatunde claims the events are mutually exclusive because “red” and “blue” are different colors. Explain the error and give a possible outcome satisfying both events.

Hints

- Identify which component of the ordered outcome each event constrains. - Different stages can carry different colors simultaneously. - Construct one shared ordered outcome.

Solution

1. The color conditions apply to different draw positions. 2. An outcome such as \((\text{red first}, \text{blue second})\) satisfies both. 3. Different labels are not mutually exclusive when they can occur at different stages.

Answer

The events are not mutually exclusive; “red first, blue second” lies in their intersection.
Mutually exclusive events 5
At a school, the probability that a randomly selected student is in Grade 11 is \(0.28\), and the probability that the student is in Grade 12 is \(0.24\). These events are mutually exclusive. What is the probability that the student is in Grade 11 or Grade 12?

Hints

- Use the fact that the two grade events cannot happen together. - For events with no overlap, the union does not require subtracting an intersection. - Combine the two given probabilities.

Solution

1. A student cannot be in both grades, so the two events have no overlap. 2. For mutually exclusive events, add the probabilities: \(0.28+0.24=0.52\).

Answer

\(0.52\)
Mutually exclusive events 6
One ball is drawn at random from the jar. Could the ball be red and have a number divisible by \(3\)? Are those events mutually exclusive? Name a ball that supports your answer.
Figure for problem 563307

Hints

- What would a counterexample to “mutually exclusive” need to satisfy? - Does a letter-number label tell you more than the ball's color alone?

Solution

1. The red ball labeled R9 is both red and numbered with a multiple of \(3\). 2. Both events can happen on the same draw, so they are not mutually exclusive.

Answer

Yes. Ball R9 meets both descriptions. The events are not mutually exclusive.
Mutually exclusive events 7
A fair six-sided die is rolled once. Let A mean “an even number” and B mean “a number greater than \(4\).” Are A and B mutually exclusive? Name an outcome that supports your answer.

Hints

- List the numbers in A and the numbers in B. - Look for a number in both lists. - Events are mutually exclusive only when no outcome belongs to both.

Solution

1. A contains the outcomes \(2,4,6\). 2. B contains \(5,6\). 3. Since both contain \(6\), the events are not mutually exclusive.

Answer

No. Rolling a \(6\) makes both events happen.
Mutually exclusive events 8
A concert-ticket draw awards either a backstage tour, a signed poster, or no prize. The probability of a tour is \(0.20\), and the probability of a poster is \(0.35\). Each ticket wins at most one prize. What is the probability of winning a tour or a poster?

Hints

- Would any single ticket be counted in both prize groups? - How could you combine two chances when one outcome cannot belong to both groups?

Solution

1. A ticket cannot win both prizes, so these events are mutually exclusive. 2. Add their probabilities: \(0.20+0.35=0.55\).

Answer

\(0.55\), or \(55\%\).
Mutually exclusive events 9
A website records the number \(X\) of failed login attempts before a successful login. Split the event \(X\ge1\) into three mutually exclusive subevents that distinguish one failure, two failures, and at least three failures. Write the union and explain why the pieces are disjoint.

Hints

- Start with the possible values contained in the larger event. - Choose category boundaries so that every allowed value enters exactly one piece. - Check both overlap and coverage.

Solution

1. Use \(E_1=\{X=1\}\), \(E_2=\{X=2\}\), and \(E_3=\{X\ge3\}\). 2. No value of \(X\) can satisfy two of these descriptions, so the events are pairwise mutually exclusive. 3. Their union is \(\{X\ge1\}\).

Answer

One valid split is \(E_1=\{X=1\}\), \(E_2=\{X=2\}\), and \(E_3=\{X\ge3\}\). These events are pairwise mutually exclusive, and \(E_1\cup E_2\cup E_3=\{X\ge1\}\).
Mutually exclusive events 10
A school game uses a fair spinner numbered \(1\) through \(8\). A small prize is given for a number in \(A=\{1,2,3,4,5\}\). A bonus prize is given for a number in \(B=\{4,5,6\}\). The game leader wants a third prize that can be won only by landing in \(A\), but **never** at the same time as the bonus prize. Which spinner results can win this third prize? What is the greatest possible probability of winning it? Explain why including another result from \(A\) would break the rule.

Hints

- Which spinner numbers trigger both existing prizes? - Which numbers qualify for the small prize without triggering the bonus? - What would happen if another small-prize number were included?

Solution

1. Results \(4\) and \(5\) belong to both \(A\) and \(B\), so they cannot qualify for the third prize. 2. The largest allowed set is \(\{1,2,3\}\), giving probability \(\frac{3}{8}\). 3. Adding any other result from \(A\) would add \(4\) or \(5\), which also wins the bonus prize.

Answer

The qualifying results are \(1,2,3\); the largest possible probability is \(\frac{3}{8}\). Adding \(4\) or \(5\) would make the prize events overlap.
Mutually exclusive events 11
A survey asks respondents to choose exactly one primary news source from TV, radio, print, or online. A second survey asks respondents to select every source they use. Are “uses TV” and “uses online” mutually exclusive in each survey? Explain how the response rule changes the answer.

Hints

- Identify what one outcome permits. - The same labels can represent different event structures under different survey rules. - Look for whether a single respondent can satisfy both events.

Solution

1. In the exactly-one survey, a respondent cannot be in both categories, so the events are mutually exclusive. 2. In the select-all survey, one respondent may use both TV and online, so the events can overlap. 3. Mutual exclusivity depends on the outcome definition, not only the category names.

Answer

They are mutually exclusive in the exactly-one survey but not necessarily in the select-all survey.
Mutually exclusive events 12
A measurement \(X\) is classified using \(A=\{X\le10\}\) and \(B=\{X\ge10\}\). Are \(A\) and \(B\) mutually exclusive? Revise one inequality so the categories become mutually exclusive while still covering all real values.

Hints

- Check boundary values explicitly. - Mutually exclusive intervals cannot share an endpoint. - Exhaustive intervals must leave no gap.

Solution

1. Both events contain \(X=10\), so their intersection is not empty. 2. Replace one event with \(B=\{X>10\}\), or replace \(A\) with \(\{X<10\}\). 3. The revised pair has no overlap and still covers every real value.

Answer

No; they overlap at \(X=10\). One valid revision is \(A=\{X\le10\}\) and \(B=\{X>10\}\).
Mutually exclusive events 13
Five distinct runners finish a race in random order. Let \(A\) be “Maya finishes first” and \(B\) be “Maya finishes last.” Show that \(A\) and \(B\) are mutually exclusive and find \(P(A\cup B)\).

Hints

- One person occupies exactly one finishing position. - Use symmetry for each endpoint event. - Add the probabilities only after checking the intersection.

Solution

1. Maya cannot occupy both first and last place in one ordering, so \(A\cap B=\varnothing\). 2. By symmetry, \(P(A)=P(B)=\frac15\). 3. Therefore \(P(A\cup B)=\frac15+\frac15=\frac25\).

Answer

The events are mutually exclusive, and \(P(A\cup B)=\frac25\).
Mutually exclusive events 14
For one soccer match, the outcomes “home win,” “draw,” and “away win” are mutually exclusive. Let \(G\) be “the home team scores at least one goal.” Which of the three result events can overlap \(G\)? Explain.

Hints

- Use possible score examples. - Events from different classification variables may overlap. - Do not transfer exclusivity from one partition to an unrelated event.

Solution

1. A home win can include a home goal. 2. A draw can include a positive tied score, so it can include a home goal. 3. An away win can also occur when the away team wins \(2\) to \(1\), so it can include a home goal. Thus \(G\) can overlap all three result events.

Answer

\(G\) can overlap home win, draw, and away win. Match-result categories are exclusive with one another, but not with the scoring event.
Mutually exclusive events 15
A wildlife record classifies each observed animal by species. Let \(A\) be “the animal is a wolf” and \(B\) be “the animal is a mammal.” Are \(A\) and \(B\) mutually exclusive? Describe their intersection and explain why the category labels being different does not settle the question.

Hints

- Ask whether one category can be contained inside the other. - Describe an outcome that would satisfy both event descriptions. - Distinct names do not necessarily mean distinct, nonoverlapping sets.

Solution

1. Every wolf is a mammal, so \(A\subseteq B\). 2. Therefore \(A\cap B=A\), which is nonempty whenever wolves are possible. 3. Different verbal labels can describe nested events rather than disjoint categories.

Answer

The events are not mutually exclusive. Their intersection is \(A\cap B=A\), because every wolf is a mammal.
Mutually exclusive events 16
Two prizes at a school fair are awarded on a single spin of a wheel. The chance of a notebook is \(0.22\), and the chance of a water bottle is \(0.31\). Each spin awards at most one prize. a) What is the probability that a spin wins **exactly one** of these prizes? b) Suppose a different fair allows both prizes on one spin. Could you still add \(0.22+0.31\) to find the probability of exactly one prize? Explain what information would be missing.

Hints

- Does a win in the first game allow both prizes on the same spin? - What does “exactly one” leave out in the second game? - Which possible outcome would cause simple addition to count too much?

Solution

1. In the first game, the two prize events cannot happen together. Winning exactly one is the same as winning either prize, so its probability is \(0.22+0.31=0.53\). 2. In the second game, a spin winning both prizes would not count as exactly one. The shared-event probability would be needed to subtract that overlap appropriately.

Answer

a) \(0.53\). b) No. The chance of winning both prizes would be needed because “exactly one” excludes spins that win both.
Mutually exclusive events 17
At a school, \(60\%\) of students play a sport. Of all students, \(25\%\) play a sport and also play an instrument. a) What percentage play a sport but **do not** play an instrument? b) Why can you add the answers for “sport and instrument” and “sport but no instrument” to get the overall sport percentage?

Hints

- Split the sport group into those who play an instrument and those who do not. - Can any student fall into both of those smaller groups? - How do the two smaller groups relate to the whole sport group?

Solution

1. Every sport-playing student either plays an instrument or does not; no student can be in both of these smaller groups. 2. The required percentage is \(60\%-25\%=35\%\). 3. The two nonoverlapping sport groups together make up every student who plays a sport, so their shares add: \(25\%+35\%=60\%\).

Answer

a) \(35\%\). b) The two groups do not overlap and together include every student who plays a sport.
Mutually exclusive events 18
A whole number is chosen at random from \(0\) through \(10\), with each number equally likely. The number line shows events \(A\) and \(B\). Both event bands include their endpoints. a) List the numbers in \(A\cap B\). b) Find \(P(A\cap B)\). c) Are \(A\) and \(B\) mutually exclusive? Explain.
Figure for problem 547292

Hints

- Which whole-number tick marks are covered by both bands? - Count all the equally likely numbers that could be chosen, including both endpoints. - If one number belongs to both events, can the events be mutually exclusive?

Solution

1. The bands overlap at the whole numbers \(4\), \(5\), and \(6\). 2. There are \(11\) equally likely numbers from \(0\) through \(10\), so \(P(A\cap B)=\frac{3}{11}\). 3. Both events can occur for three outcomes, so they are not mutually exclusive.

Answer

a) \(A\cap B=\{4,5,6\}\). b) \(P(A\cap B)=\frac{3}{11}\). c) No. The events overlap at \(4\), \(5\), and \(6\).
Mutually exclusive events 19
A teacher makes \(12\) equally likely raffle tickets numbered \(1\) to \(12\). A ticket can win a notebook, a pen, or neither, but cannot win both prizes. The desired probabilities are \(\frac{1}{3}\) for a notebook and \(\frac{1}{4}\) for a pen. Choose specific ticket numbers for the two prizes. How many tickets win neither prize? Explain why your assignment gives the desired chances without allowing both prizes.

Hints

- How many of the \(12\) tickets should win each prize? - Can the same ticket number appear in both winning groups? - Which ticket numbers remain after both prize groups are assigned?

Solution

1. A notebook needs \(\frac{1}{3}\cdot12=4\) tickets and a pen needs \(\frac{1}{4}\cdot12=3\) tickets. 2. One assignment is notebook on \(1,2,3,4\) and pen on \(5,6,7\). 3. The two sets do not overlap, so \(12-4-3=5\) tickets win neither prize.

Answer

For example, notebook: \(\{1,2,3,4\}\); pen: \(\{5,6,7\}\); neither: \(5\) tickets. The events are mutually exclusive and have probabilities \(\frac{4}{12}=\frac{1}{3}\) and \(\frac{3}{12}=\frac{1}{4}\).
Mutually exclusive events 20
Mutual exclusivity is not transitive. In the sample space \(S=\{1,2,3,4\}\), let \(A=\{1,2\}\), \(B=\{3\}\), and \(C=\{2,4\}\). Verify that \(A\) is mutually exclusive with \(B\) and that \(B\) is mutually exclusive with \(C\), but \(A\) is not mutually exclusive with \(C\). Explain what this counterexample disproves.

Hints

- Check each of the three pairwise intersections separately. - A transitive relation would force the first and third events to have the relation. - One shared outcome is enough to refute mutual exclusivity.

Solution

1. \(A\cap B=\varnothing\), so \(A\) and \(B\) are mutually exclusive. 2. \(B\cap C=\varnothing\), so \(B\) and \(C\) are mutually exclusive. 3. \(A\cap C=\{2\}\ne\varnothing\), so \(A\) and \(C\) are not mutually exclusive. 4. Therefore, mutual exclusivity does not satisfy the transitive property.

Answer

\(A\cap B=\varnothing\) and \(B\cap C=\varnothing\), but \(A\cap C=\{2\}\). Thus mutual exclusivity is not transitive.
Mutually exclusive events 21
Two fair dice are rolled. Let \(A\) be “the minimum is \(2\)” and \(B\) be “the maximum is \(5\).” Are \(A\) and \(B\) mutually exclusive? List their intersection.

Hints

- Translate minimum and maximum conditions into ordered outcomes. - Both conditions can determine the two values simultaneously. - Include both possible orders.

Solution

1. Both conditions hold when the two results are \(2\) and \(5\) in either order. 2. \(A\cap B=\{(2, 5),(5, 2)\}\). 3. Therefore the events are not mutually exclusive.

Answer

No. Their intersection is \(\{(2, 5),(5, 2)\}\).
Mutually exclusive events 22
A sample space has \(7\) elementary outcomes. What is the greatest number of nonempty events that can be pairwise mutually exclusive? Prove the bound and describe a family that attains it.

Hints

- Assign at least one distinct outcome to every nonempty event. - Use the fact that disjoint events cannot share an outcome. - Test the family of singleton events.

Solution

1. Each nonempty event in a pairwise disjoint family must contain at least one elementary outcome. 2. No elementary outcome can belong to two events in the family. 3. With only \(7\) outcomes, there can be at most \(7\) nonempty pairwise disjoint events. 4. The seven singleton events attain the bound.

Answer

The maximum is \(7\), attained by the seven singleton events.
Mutually exclusive events 23
Give an example of two events that are mutually exclusive but not exhaustive, and two events that are exhaustive but not mutually exclusive.

Hints

- Construct one pair with no overlap but an incomplete union. - Construct another pair whose union is full but whose intersection is nonempty. - A fair die offers a small sample space for checking both properties.

Solution

1. On a die, “roll \(1\)” and “roll \(2\)” are mutually exclusive but do not cover all outcomes. 2. “Roll even” and “roll at most \(5\)” are exhaustive because outcomes \(1\) through \(5\) satisfy the second event and outcome \(6\) satisfies the first. They overlap at \(2\) and \(4\). 3. Thus the two properties are logically distinct.

Answer

Example: \(\{1\}\) and \(\{2\}\) are mutually exclusive but not exhaustive. “Roll even” and “roll at most \(5\)” are exhaustive but not mutually exclusive.
Mutually exclusive events 24
A school alarm has two detectors: smoke and heat. During one check, neither, either one, or both detectors may sound. Let \(A\) mean **both** detectors sound. Let \(B\) mean **at least one** detector does not sound. a) List the detector results that belong to \(B\). b) Can \(A\) and \(B\) occur together? Do they cover every possible result? Explain.

Hints

- List the four ways two detectors can sound or stay silent. - Which results meet “at least one does not sound”? - Is any result shared by both events, or left out of both?

Solution

1. The possible results are neither sounds, only smoke sounds, only heat sounds, or both sound. 2. Event \(B\) includes the first three results because at least one detector is silent; event \(A\) includes only the both-sound result. 3. These events cannot happen together and cover all four results, so they are mutually exclusive and exhaustive.

Answer

a) Neither sounds, only smoke sounds, and only heat sounds. b) No, they cannot happen together. Yes, they cover all possible results; one of the events occurs on every check.
Mutually exclusive events 25
A fair six-sided die is rolled. Let \(A\) be the event “the result is even” and \(B\) be the event “the result is greater than \(4\).” Jordan says, “\(P(A\cup B)=P(A)+P(B)=\frac12+\frac13=\frac56\).” Explain the error and find the correct value of \(P(A\cup B)\).

Hints

- List the outcomes in each event before deciding whether the events are mutually exclusive. - Check whether any die result belongs to both sets. - A shared outcome must be counted only once in the union.

Solution

1. \(A=\{2,4,6\}\) and \(B=\{5,6\}\). 2. The events are not mutually exclusive because outcome \(6\) is in both events. 3. The union is \(\{2,4,5,6\}\), so \(P(A\cup B)=\frac46=\frac23\). 4. Equivalently, subtract the overlap once: \(\frac12+\frac13-\frac16=\frac23\).

Answer

The events overlap at outcome \(6\), so their probabilities cannot simply be added. The correct probability is \(\frac23\).
Mutually exclusive events 26
At a neighborhood diner, each lunch order has one main dish: pasta, a chicken sandwich, or something else. The chance of pasta or a chicken sandwich is \(0.70\). The chance of pasta is \(0.25\). What is the chance of a chicken sandwich?

Hints

- Could a single order belong to both named main-dish groups? - Would the combined probability be larger than either separate probability? - Which part of the combined event has not yet been accounted for?

Solution

1. An order has one main dish, so pasta and chicken-sandwich orders do not overlap. 2. Subtract the pasta chance from the combined chance: \(0.70-0.25=0.45\).

Answer

\(0.45\), or \(45\%\).
Mutually exclusive events 27
The numbered dots are equally likely outcomes. Milo adds the probability of A to the probability of B and gets \(1\). Explain his mistake, then find the probability of A or B.
Figure for problem 563311

Hints

- Could one dot be named in both groups? - How would counting all eligible dots differ from adding the two group counts? - Check whether your final probability matches the number of distinct outcomes in the circles.

Solution

1. Circle A contains outcomes 1, 2, 3, and 4; circle B contains 3, 4, 5, and 6. 2. Outcomes 3 and 4 are in both circles, so adding \(\frac{4}{8}+\frac{4}{8}\) counts them twice. 3. The outcomes in A or B are 1 through 6, so \(P(A\cup B)=\frac{6}{8}=\frac{3}{4}\).

Answer

Milo counted outcomes 3 and 4 twice. The correct probability of A or B is \(\frac{6}{8}=\frac{3}{4}\).
Mutually exclusive events 28
A fair six-sided die is rolled once. Event A is “a number less than \(3\),” and event B is “a number greater than \(4\).” a) Are A and B mutually exclusive? b) Can the die show a number that belongs to neither event? Give an example.

Hints

- List the die faces that satisfy each description. - Look for any number that could satisfy both descriptions. - Then check whether any face satisfies neither description.

Solution

1. A consists of 1 and 2, while B consists of 5 and 6. No number belongs to both, so they are mutually exclusive. 2. A roll of 3 or 4 belongs to neither event, so the events do not cover every possible outcome.

Answer

a) Yes. A and B have no outcomes in common. b) Yes. A roll of \(3\) or \(4\) belongs to neither.
Mutually exclusive events 29
Event \(A\) has probability \(0.62\). Event \(B\) must be mutually exclusive with \(A\). a) What is the greatest possible value of \(P(B)\)? b) Under what additional condition is that greatest value attained?

Hints

- Where can event \(B\) occur if no outcome may belong to both events? - Compare \(B\) with the complement of \(A\). - Think about what must happen for the available probability outside \(A\) to be used completely.

Solution

1. Since \(A\cap B=\varnothing\), event \(B\) must lie within \(A^c\). 2. \(P(A^c)=1-0.62=0.38\), so \(P(B)\le0.38\). 3. Equality occurs when \(B=A^c\), so the two events are also exhaustive.

Answer

a) The greatest possible value is \(0.38\). b) It is attained when \(B=A^c\), making \(A\) and \(B\) exhaustive as well as mutually exclusive.
Mutually exclusive events 30
Events \(A,B,C\) form a partition of the sample space, and each has positive probability. a) Find \(A^c\cap B^c\) in terms of the partition. b) Are \(A^c\) and \(B^c\) mutually exclusive? c) Generalize the conclusion to complements of two distinct cells of a partition with at least three nonempty cells.

Hints

- Describe what remains after excluding each of two partition categories. - Use both disjointness and exhaustiveness of the original partition. - Consider where a third category lies relative to the two complements.

Solution

1. Because \(A,B,C\) are disjoint and exhaustive, outcomes outside both \(A\) and \(B\) are exactly the outcomes in \(C\). 2. Thus \(A^c\cap B^c=C\). 3. Since \(P(C)>0\), the complements overlap and are not mutually exclusive. 4. In any partition with at least three nonempty cells, the complements of two cells share every other cell.

Answer

a) \(A^c\cap B^c=C\) b) No, because their intersection has positive probability. c) The complements of two partition cells overlap on all remaining nonempty cells.
Mutually exclusive events 31
Suppose \(P(A)=0.72\) and \(P(B)=0.46\). Prove that \(A\) and \(B\) cannot be mutually exclusive, and find the smallest possible value of \(P(A\cap B)\).

Hints

- A union probability cannot exceed \(1\). - Use the general addition rule rather than the disjoint version. - Maximize the union to minimize the overlap.

Solution

1. If the events were mutually exclusive, their union would have probability \(0.72+0.46=1.18\), which is impossible. 2. The addition rule gives \(P(A\cap B)=P(A)+P(B)-P(A\cup B)\). 3. Since \(P(A\cup B)\le1\), the intersection is at least \(1.18-1=0.18\).

Answer

They cannot be mutually exclusive. The smallest possible intersection probability is \(0.18\).
Mutually exclusive events 32
In a survey of \(100\) students, \(40\) play soccer, \(30\) play basketball, and \(25\) play in the band. Of the basketball players, \(10\) also play soccer. Of the band members, \(8\) also play soccer, \(7\) also play basketball, and \(3\) do **all three** activities. Count students in three groups: (1) soccer players; (2) basketball players who do not play soccer; (3) band members who do neither sport. Why can these counts be added without counting anyone twice? How many students do at least one activity?

Hints

- Count everyone in the first activity before finding who is new in the second. - How many band members are already counted in at least one sport? - Check whether a student can belong to more than one of the three groups as defined.

Solution

1. Group 1 contains \(40\) soccer players. 2. Group 2 contains \(30-10=20\) basketball players who are not soccer players. 3. Among the band, \(8+7-3=12\) play at least one of the sports, so group 3 contains \(25-12=13\). 4. The three groups cannot overlap and together include every student in at least one activity. The total is \(40+20+13=73\).

Answer

Soccer: \(40\); basketball but not soccer: \(20\); band but neither sport: \(13\). The groups do not overlap, and \(73\) students do at least one activity.
Mutually exclusive events 33
In sample space \(S=\{1,2,3,4,5,6\}\), let \( A=\{1,2\},\quad B=\{3,4\},\quad C=\{1,3\}. \) The triple intersection is empty. Are the three events pairwise mutually exclusive? Explain.

Hints

- Check every pair, not only the intersection of all three. - Pairwise mutual exclusivity requires three separate empty intersections. - A triple intersection can be empty even when pairs overlap.

Solution

1. \(A\cap B=\varnothing\). 2. \(A\cap C=\{1\}\) and \(B\cap C=\{3\}\). 3. An empty triple intersection does not imply pairwise mutual exclusivity; two pairs overlap.

Answer

No. Only \(A\) and \(B\) are mutually exclusive; \(C\) overlaps both.
Mutually exclusive events 34
In a sample of \(500\) retail transactions, no transaction was observed to use both curbside pickup and cash payment. Does the zero observed intersection prove that the two population events are mutually exclusive? Distinguish a structural zero from a sample zero.

Hints

- Separate what happened in this finite sample from what is logically possible in the population. - Ask what additional information would be needed to establish impossibility. - A zero count and an impossible category combination are different claims.

Solution

1. The observed count of zero says only that the sampled transactions contained no overlap. 2. Mutual exclusivity requires the two events to be unable to occur together in the population. 3. The zero would be structural only if the transaction definitions or store rules made the combination impossible. Without such a restriction, it is only a sample zero and does not prove mutual exclusivity.

Answer

No. An observed zero is a sample zero unless the event definitions or rules make the overlap impossible. Mutual exclusivity requires a structural impossibility, not merely zero observed overlap in one sample.
Mutually exclusive events 35
A probability space contains pairwise mutually exclusive events \(E_1,\ldots,E_k\), each having probability at least \(0.22\). What is the greatest possible value of \(k\)? Show that the bound can be attained by giving possible event probabilities.

Hints

- Add the lower bounds for all disjoint events. - Compare the resulting sum with the maximum possible probability. - Check that an example reaches the integer bound.

Solution

1. Pairwise mutually exclusive probabilities add, so \(0.22k\le1\). 2. This gives \(k\le4.545\ldots\), hence \(k\le4\). 3. Four events are possible, for example with probabilities \(0.22,0.22,0.22,\) and \(0.34\). 4. Their probabilities sum to \(1\), so the maximum is \(4\).

Answer

The greatest possible value is \(k=4\). One attainable set of probabilities is \(0.22,0.22,0.22,0.34\).
Mutually exclusive events 36
Three yes/no attributes \(A,B,\) and \(C\) may overlap in any way. What is the smallest standard set of mutually exclusive categories that preserves every possible response pattern? List the categories symbolically and state how many there are.

Hints

- Count the possible yes/no patterns for three attributes. - Represent each pattern with an intersection of events or complements. - Check that every response has exactly one pattern.

Solution

1. Each attribute can be present or absent, giving \(2^3=8\) truth-value patterns. 2. The categories are \(A\cap B\cap C\), \(A\cap B\cap C^c\), \(A\cap B^c\cap C\), \(A\cap B^c\cap C^c\), \(A^c\cap B\cap C\), \(A^c\cap B\cap C^c\), \(A^c\cap B^c\cap C\), and \(A^c\cap B^c\cap C^c\). 3. Each response pattern belongs to exactly one of these categories, so they are pairwise disjoint and exhaustive. 4. No coarser standard scheme preserves all eight distinct yes/no patterns.

Answer

The \(8\) categories are \(A\cap B\cap C\), \(A\cap B\cap C^c\), \(A\cap B^c\cap C\), \(A\cap B^c\cap C^c\), \(A^c\cap B\cap C\), \(A^c\cap B\cap C^c\), \(A^c\cap B^c\cap C\), and \(A^c\cap B^c\cap C^c\).
Mutually exclusive events 37
Sofia claims that if \(P(A)+P(B)=1\), then \(A\) and \(B\) must be mutually exclusive and exhaustive. The diagram shows the four regions determined by \(A\) and \(B\) inside the sample space. Let \(P(A)=0.60\), \(P(B)=0.40\), and \(P(A\cap B)=0.15\). Find the probability of each region and explain why Sofia's conclusion fails.
Figure for problem 547300

Hints

- Start with the given overlap and subtract it from each marginal probability. - The outcomes outside both circles form the complement of the union. - Check whether the overlap and neither regions are actually zero.

Solution

1. The overlap has probability \(P(A\cap B)=0.15\). 2. The A-only region has probability \(0.60-0.15=0.45\), and the B-only region has probability \(0.40-0.15=0.25\). 3. The addition rule gives \(P(A\cup B)=0.60+0.40-0.15=0.85\), so the neither region has probability \(1-0.85=0.15\). 4. Because both the overlap and neither regions have positive probability, the events are neither mutually exclusive nor exhaustive even though their marginal probabilities sum to \(1\).

Answer

A only: \(0.45\) Both: \(0.15\) B only: \(0.25\) Neither: \(0.15\) The events are neither mutually exclusive nor exhaustive.
Mutually exclusive events 38
Events \(A,B,C\) form a partition of a sample space. Event \(B\) is then split into two events \(B_1\) and \(B_2\) that are mutually exclusive and satisfy \(B_1\cup B_2=B\). Prove that \(A,B_1,B_2,C\) also form a partition.

Hints

- Use the fact that both new events stay inside the original event being split. - Check pairwise overlap separately from total coverage. - Replace the union of the two new pieces with the original event.

Solution

1. Because \(B_1,B_2\subseteq B\), each is disjoint from \(A\) and \(C\), which were disjoint from \(B\). 2. \(B_1\) and \(B_2\) are disjoint by assumption. 3. The union is \(A\cup B_1\cup B_2\cup C=A\cup B\cup C=S\). 4. The four events are pairwise mutually exclusive and exhaustive, so they form a partition.

Answer

The four events form a partition: they are pairwise mutually exclusive, and \(A\cup B_1\cup B_2\cup C=A\cup B\cup C=S\).
Mutually exclusive events 39
For arbitrary events \(A\) and \(B\), show that the following statements are equivalent: 1. \(A\) and \(B\) are mutually exclusive. 2. \(A\subseteq B^c\). Explain both directions using outcome membership.

Hints

- Translate mutual exclusivity into a statement about individual outcomes. - What does membership in a complement mean? - Prove each implication separately rather than assuming they are reversible.

Solution

1. If \(A\cap B=\varnothing\), then no outcome in \(A\) belongs to \(B\); therefore every outcome in \(A\) belongs to \(B^c\), so \(A\subseteq B^c\). 2. If \(A\subseteq B^c\), an outcome in \(A\) cannot also be in \(B\); therefore \(A\cap B=\varnothing\). 3. The two statements are equivalent.

Answer

\(A\cap B=\varnothing\) if and only if \(A\subseteq B^c\). The first condition says no outcome lies in both events, which is exactly the requirement that every outcome of \(A\) lie outside \(B\).
Mutually exclusive events 40
A commuter selects exactly one primary route type: highway, local roads, or rail. Event \(R\) is “primary route is rail.” Event \(T\) is “uses a train at some point,” including park-and-ride trips. Are \(R\) and \(T^c\) mutually exclusive? Are “highway” and \(T\) mutually exclusive?

Hints

- Use the logical implication built into the primary-rail category. - Events from primary and secondary travel descriptions need not form one partition. - Test each pair separately.

Solution

1. If the primary route is rail, the commuter uses a train, so \(R\cap T^c=\varnothing\). 2. A highway-primary commuter may use a train for part of the trip under the stated definition. 3. Therefore highway and \(T\) are not necessarily mutually exclusive.

Answer

\(R\) and \(T^c\) are mutually exclusive. Highway and \(T\) may overlap.
Mutually exclusive events 41
A finite sample space is \(S=\{s_1,\ldots,s_n\}\), and every elementary outcome has positive probability. Events \(A\) and \(B\) satisfy \(P(A\cap B)=0\). Prove that \(A\) and \(B\) are mutually exclusive. Then explain why the positive-probability assumption on every elementary outcome is essential.

Hints

- Use contradiction: suppose the intersection contains one elementary outcome. - Compare the probability of a set with the probability of a positive-probability subset it contains. - Distinguish an empty event from a nonempty null event.

Solution

1. If \(A\cap B\) contained an elementary outcome \(s_i\), then \(P(A\cap B)\ge P(\{s_i\})>0\), contradicting \(P(A\cap B)=0\). 2. Therefore \(A\cap B=\varnothing\), so the events are mutually exclusive. 3. Without the positive-atom assumption, a nonempty event may have probability \(0\), as can occur for single points in a continuous model. Then zero intersection probability would not force an empty intersection.

Answer

Under the stated finite positive-atom condition, \(A\cap B=\varnothing\), so \(A\) and \(B\) are mutually exclusive. The conclusion can fail when nonempty events of probability \(0\) are allowed.
Mutually exclusive events 42
Events \(A\) and \(B\) are mutually exclusive, and \(C\) is any event. Prove that \((A\cup C)\cap(B\cup C)=C\). Explain why adjoining the same event \(C\) to two disjoint events makes their new intersection exactly \(C\).

Hints

- What would you need to establish to prove that the two displayed sets are equal? - Start with an arbitrary outcome that belongs to both \(A\cup C\) and \(B\cup C\). What possibilities are there for that outcome? - If such an outcome were not in \(C\), what would the two union memberships force, and how would that interact with mutual exclusivity?

Solution

1. Distribute intersection over union: \((A\cup C)\cap(B\cup C)=(A\cap B)\cup(A\cap C)\cup(C\cap B)\cup C\). 2. Since \(A\cap B=\varnothing\), the first term disappears. 3. Both \(A\cap C\) and \(B\cap C\) are subsets of \(C\), so their union with \(C\) is simply \(C\).

Answer

\((A\cup C)\cap(B\cup C)=C\).

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