Mutually Exclusive Events
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AP Statistics › Mutually Exclusive Events
A hospital records one patient’s status at a randomly chosen time today. Let event $A$ be “the patient is in the emergency department” and event $B$ be “the patient is in the operating room.” Hospital policy states a patient can be in only one location at a time. Are the two events mutually exclusive?
Yes; the events are mutually exclusive because the patient cannot be in two locations at the same time.
No; the events are independent because location changes over time.
Cannot be determined without knowing how long the patient stayed in each location.
Yes; the events are mutually exclusive only if $P(A)+P(B)=1$.
No; the events are not mutually exclusive because a patient could be in both places during the same time.
Explanation
This hospital location question perfectly demonstrates mutually exclusive events. The key constraint is that "a patient can be in only one location at a time." Since a patient cannot physically be in both the emergency department AND the operating room simultaneously, these events are mutually exclusive. When one event occurs, the other cannot occur at that same randomly chosen time. Choice A incorrectly suggests a patient could be in two places at once, which violates basic physics and the stated hospital policy. This is a real-world application of the mutual exclusivity concept.
A number is randomly selected from the set ${1,2,3,4,5,6,7,8,9,10}$. Let event $A$ be “the number is prime” and event $B$ be “the number is even.” The number 2 is both prime and even, so the events can occur together. Are the two events mutually exclusive?
No, because 2 is in both events, so $A \cap B \neq \varnothing$.
Yes, because selecting one number prevents any overlap.
Yes, because the events are not independent.
No, because mutually exclusive events must have $P(A)=P(B)$.
Yes, because primes are odd and evens are not.
Explanation
This question tests recognition of mutually exclusive events in AP Statistics, where events lack shared outcomes, making simultaneous occurrence impossible. Events A (prime) and B (even) are not mutually exclusive because 2 is both prime and even, resulting in a non-empty $A \cap B$. Choice B properly explains no, due to the overlap at 2. A distractor such as choice A might attract those forgetting 2 is prime, incorrectly separating primes and evens completely. For a mini-lesson, mutual exclusivity means the intersection is empty; if not, events overlap, and you must account for $P(A \cap B)$ when finding union probabilities.
A student randomly selects one card from a standard 52-card deck and records the outcome. Let event $A$ be “the card is a heart” and event $B$ be “the card is a king.” A king of hearts exists in the deck, so the two events can occur on the same draw. Are the two events mutually exclusive?
Yes, because hearts and kings are different categories of cards.
No, because the king of hearts makes $A$ and $B$ able to occur together.
Yes, because $P(A\cap B)=0$ for a single draw from a deck.
No, because the events are independent.
Yes, because knowing the card is a heart makes it less likely to be a king.
Explanation
This question assesses the concept of mutually exclusive events in AP Statistics, where two events are mutually exclusive if they cannot occur at the same time, meaning their intersection is empty and P(A ∩ B) = 0. Here, events A (drawing a heart) and B (drawing a king) are not mutually exclusive because the king of hearts is both a heart and a king, so there is an outcome in A ∩ B. The correct answer is no, they are not mutually exclusive, as indicated by choice C. A common distractor is choice A, which incorrectly assumes P(A ∩ B) = 0 for a single draw, but in reality, it is not zero due to the king of hearts. As a mini-lesson, remember that mutual exclusivity focuses on whether events can overlap in the sample space; if any outcome satisfies both, they are not mutually exclusive. Independence is a separate concept and does not determine mutual exclusivity.
A bag contains only chocolate, vanilla, and strawberry candies. One candy is selected at random. Define event $A$ as “the candy is chocolate” and event $B$ as “the candy is not chocolate.” The candy selected must be exactly one flavor. Are the two events mutually exclusive?
No; the events are independent because $B$ is the complement of $A$.
Cannot be determined without knowing the number of each flavor.
No; the events are not mutually exclusive because some candies could be mixed flavors.
Yes; the events are mutually exclusive because a candy cannot be both chocolate and not chocolate.
Yes; the events are mutually exclusive only if chocolate is the most common flavor.
Explanation
This candy selection problem involves complementary events. Event A is "chocolate" and event B is "not chocolate," making B the complement of A. By definition, complementary events are always mutually exclusive because an outcome cannot simultaneously be both in a set and not in that set. A candy cannot be both chocolate and not chocolate at the same time. Since every candy must be either chocolate or not chocolate (but not both), these events perfectly demonstrate mutual exclusivity. Choice B incorrectly introduces the idea of mixed flavors, but the problem states each candy is exactly one flavor.
A number is selected at random from the integers $1$ through $20$, inclusive. Define event $A$ as “the number is even” and event $B$ as “the number is a multiple of 5.” Are the two events mutually exclusive?
Yes; the events are mutually exclusive because they involve different properties.
No; the events are independent, so they cannot be mutually exclusive.
Cannot be determined without counting how many evens and multiples of 5 there are.
Yes; no number from 1 to 20 can be both even and a multiple of 5.
No; the events are not mutually exclusive because some numbers are both even and multiples of 5.
Explanation
This problem asks about the mutual exclusivity of selecting an even number versus a multiple of 5 from 1-20. To determine this, we need to check if any numbers satisfy both conditions. The numbers 10 and 20 are both even AND multiples of 5, meaning both events can occur with the same outcome. Since there exist outcomes where both events happen simultaneously, the events are not mutually exclusive. Choice A incorrectly claims no overlap exists, while choice D confuses independence with mutual exclusivity. Remember: mutually exclusive events have no outcomes in common.
A school cafeteria randomly selects 1 student ID from all students who ate lunch today. Define event $A$ as “the selected student bought pizza” and event $B$ as “the selected student bought a salad.” Each student bought exactly one main dish (either pizza, salad, or a sandwich), and no student bought more than one main dish. Are the two events mutually exclusive?
No; the events are independent because buying pizza does not affect buying a salad.
Cannot be determined without knowing how many students bought each item.
No; the events are not mutually exclusive because some students could have bought both pizza and salad.
Yes; the events are mutually exclusive only if $P(A)=P(B)$.
Yes; the events are mutually exclusive because a student cannot buy both pizza and salad as their one main dish.
Explanation
This question tests understanding of mutually exclusive events in the context of cafeteria purchases. Since each student bought exactly one main dish (pizza, salad, or sandwich), a student cannot have bought both pizza AND salad. When two events cannot occur simultaneously, they are mutually exclusive. The key phrase "exactly one main dish" ensures no overlap between events A and B. Choice A incorrectly confuses independence with mutual exclusivity, while choice B misunderstands the constraint that each student bought only one main dish.
A weather app records tomorrow’s forecast for a city. Define event $A$ as “tomorrow’s forecast includes rain” and event $B$ as “tomorrow’s forecast includes thunderstorms.” The app allows multiple conditions to be listed for the same day. Are the two events mutually exclusive?
Cannot be determined without knowing the probability of thunderstorms.
No; the events are not mutually exclusive because a forecast can include both rain and thunderstorms.
Yes; thunderstorms and rain cannot occur on the same day.
No; the events are independent, so they cannot be mutually exclusive.
Yes; the events are mutually exclusive because they are different weather types.
Explanation
This weather forecast question examines whether rain and thunderstorms are mutually exclusive events. The key detail is that the app "allows multiple conditions to be listed for the same day." In meteorology, thunderstorms typically involve rain, so a forecast can include both conditions simultaneously. Since both events can occur together (a rainy day with thunderstorms), they are not mutually exclusive. Choice A incorrectly assumes these weather conditions cannot coexist, while choice D confuses independence with mutual exclusivity. Mutually exclusive events have P(A and B) = 0, which is not true here.
A standard 52-card deck is thoroughly shuffled, and 1 card is drawn at random. Let event $A$ be “the card is a heart” and event $B$ be “the card is a face card (J, Q, or K).” Are the two events mutually exclusive?
Yes; the events are mutually exclusive because suits and ranks are independent.
Cannot be determined without knowing the probability of each event.
Yes; a card cannot be both a heart and a face card.
No; the events are not mutually exclusive because there are heart face cards.
No; the events are independent, so they are not mutually exclusive.
Explanation
This question examines whether drawing a heart and drawing a face card are mutually exclusive events. In a standard deck, there are face cards (Jack, Queen, King) in every suit, including hearts. Specifically, the Jack of Hearts, Queen of Hearts, and King of Hearts are cards that satisfy both conditions. Since these cards exist, a single draw can result in both events occurring simultaneously. Events are mutually exclusive only when they cannot happen together, which is not the case here. The existence of heart face cards means these events have a non-empty intersection.
A spinner is divided into 8 equal sections labeled 1 through 8, and it is spun once. Define event $A$ as “the result is 3” and event $B$ as “the result is an odd number.” Are the two events mutually exclusive?
No; the events are not mutually exclusive because 3 is odd.
Cannot be determined without knowing the spinner’s probabilities.
Yes; the events are mutually exclusive because one event is a single number and the other is a set.
No; the events are independent, so they cannot be mutually exclusive.
Yes; the events are mutually exclusive because 3 is not odd.
Explanation
This spinner problem asks whether landing on 3 and landing on an odd number are mutually exclusive. Since 3 is an odd number (not divisible by 2), when the spinner lands on 3, both events occur simultaneously. The event "result is 3" is actually a subset of the event "result is odd," so they have a non-empty intersection. Events are only mutually exclusive when they cannot occur together, but here they can and do overlap. Choice A incorrectly claims 3 is not odd, which is mathematically false. Remember: if one event is contained within another, they cannot be mutually exclusive.
A jar contains only red, blue, and green marbles. One marble is drawn at random. Let event $A$ be “the marble is red” and event $B$ be “the marble is blue.” Because the jar contains only these colors, the marble drawn can be exactly one color. Are the two events mutually exclusive?
Yes; the events are mutually exclusive because one marble cannot be both red and blue.
No; the events are not mutually exclusive because a marble could be both red and blue.
Yes; the events are mutually exclusive only if the numbers of red and blue marbles are equal.
No; the events are independent because the jar has multiple colors.
Cannot be determined without knowing the total number of marbles.
Explanation
This question tests the fundamental concept of mutually exclusive events using marble colors. Since each marble has exactly one color and the events are "red" and "blue," a single marble cannot be both red and blue simultaneously. This is a classic example of mutually exclusive events - when one occurs, the other cannot. The fact that there are also green marbles doesn't affect the mutual exclusivity of red and blue. Choice A incorrectly suggests a marble could have multiple colors, which contradicts the given information that each marble is exactly one color.