- A spinner has 8 equal sectors labeled 1 through 8. The spinner is spun once. Let Event A be "the result is even," and Event B be "the result is greater than 6." Which statement about Events A and B is correct?
Answer: Events A and B are independent if and only if P(A∩B)=P(A)P(B).
This question requires checking independence for spinner outcomes. Event A (even result) includes {2,4,6,8} with P(A) = 4/8 = 1/2. Event B (greater than 6) includes {7,8} with P(B) = 2/8 = 1/4. The intersection A∩B contains only {8}, so P(A∩B) = 1/8. To verify independence: P(A)P(B) = (1/2)(1/4) = 1/8 = P(A∩B). The multiplication rule holds, confirming the events are independent. Choice C correctly states this definition. Choice A gives a vague reason. Choice B incorrectly applies the addition rule for disjoint events. Choice D incorrectly claims overlapping events are disjoint. The key insight is that we must check whether P(A∩B) = P(A)P(B). - A bag contains 5 white and 5 black chips. Two chips are drawn with replacement. Let Event A be "the first chip is white" and Event B be "the second chip is white." Which statement about Events A and B is correct?
Answer: Events A and B are independent because replacing the first chip keeps the second-draw probabilities the same.
This question contrasts sampling with and without replacement. When drawing with replacement, each draw is from the same distribution: P(white) = 5/10 = 1/2 for both draws. Events A and B are independent because P(B|A) = P(B|A') = P(B) = 1/2 - the first draw's outcome doesn't affect the second draw's probabilities since the chip is replaced. We can verify: P(A∩B) = P(A)·P(B) = (1/2)·(1/2) = 1/4. The replacement ensures that each draw is independent, maintaining the same probability distribution throughout the experiment.
- In a town, 55% of adults own a car, 35% own a bicycle, and 20% own both. Let Event A be "randomly selected adult owns a car" and Event B be "randomly selected adult owns a bicycle." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(A∩B)>P(A)P(B).
This question evaluates understanding of independent events by comparing P(A ∩ B) to P(A)P(B) in a survey of adult ownership. With P(A) = 0.55 for owning a car, P(B) = 0.35 for owning a bicycle, and P(A ∩ B) = 0.20, while 0.55 * 0.35 = 0.1925, which is less than 0.20, indicating positive dependence (not independent). The relationship suggests owning one increases the likelihood of owning the other. Distractor choice A confuses the independence test with the addition rule for unions. Choice C wrongly assumes mutual exclusivity from the context. Mini-lesson: Independence requires P(A ∩ B) = P(A)P(B), meaning no influence between events; mutual exclusivity means P(A ∩ B) = 0, and the two concepts differ because exclusive events are dependent if both are possible. - A card is drawn at random from a standard 52-card deck. Let Event A be "the card is a heart" and Event B be "the card is a face card (J, Q, or K)." Which statement about Events A and B is correct?
Answer: Events A and B are independent because P(A∩B)=P(A)P(B).
This classic card problem tests independence of suit and rank. In a standard deck: P(A) = 13/52 = 1/4 (13 hearts), P(B) = 12/52 = 3/13 (12 face cards total), and P(A∩B) = 3/52 (3 face cards in hearts: J♥, Q♥, K♥). Checking independence: P(A)·P(B) = (1/4)·(3/13) = 3/52 = P(A∩B). Since the equation holds, events A and B are independent. Choice B incorrectly claims hearts and face cards cannot occur together, but the Jack, Queen, and King of hearts are counterexamples. In a well-shuffled deck, knowing a card's suit doesn't change the probability it's a face card. - A company has 80 employees. Of these, 30 work remotely (R), 28 are in the sales department (S), and 12 are both remote and in sales. One employee is selected at random. Let Event A be "the employee works remotely" and Event B be "the employee is in sales." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(A∩B)=P(A)P(B).
This question involves checking independence using given frequencies. From the data: P(A) = 30/80, P(B) = 28/80, and P(A∩B) = 12/80. To verify independence, we check if P(A∩B) = P(A)·P(B). Calculating: P(A)·P(B) = (30/80)·(28/80) = 840/6400 = 21/160, while P(A∩B) = 12/80 = 24/160. Since 21/160 ≠ 24/160, the events are not independent. The fact that 12 employees are in both categories (choice A) doesn't determine independence—we must check if the probability relationship holds. Events that overlap can still be independent if they satisfy the multiplication rule. - In a town, 40% of households have a dog, 30% have a cat, and 10% have both a dog and a cat. A household is selected at random. Let Event A be "has a dog," and Event B be "has a cat." Which statement about Events A and B is correct?
Answer: Events A and B are independent if and only if P(A∩B)=P(A)P(B).
This question provides probabilities to test for independence. Given: P(A) = 0.40, P(B) = 0.30, and P(A∩B) = 0.10. To check independence, we calculate P(A)P(B) = (0.40)(0.30) = 0.12. Since P(A∩B) = 0.10 ≠ 0.12 = P(A)P(B), the multiplication rule fails, so events A and B are not independent. Choice C correctly identifies the multiplication rule as the test for independence. The fact that 10% have both pets shows the events overlap and are not disjoint. Choice B incorrectly equates P(A∩B) with P(A)+P(B). The key lesson is that we must verify the multiplication rule holds, not just assume events are independent. - A jar contains 25 candies. Of these, 15 are chocolate (C), 10 are fruit (not chocolate), and 8 candies are individually wrapped (W). Among the 15 chocolate candies, 3 are wrapped; among the 10 fruit candies, 5 are wrapped. One candy is selected at random. Let Event A be "the candy is chocolate" and Event B be "the candy is wrapped." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(A∩B)=P(A)P(B).
This problem involves calculating independence with overlapping categories. From the given information: P(A) = 15/25, P(B) = 8/25, and P(A∩B) = 3/25 (wrapped chocolate candies). To check independence: P(A)·P(B) = (15/25)·(8/25) = 120/625 = 24/125, while P(A∩B) = 3/25 = 15/125. Since 24/125 ≠ 15/125, events A and B are not independent. Choice E incorrectly claims chocolate and wrapped candies cannot overlap, but the problem explicitly states 3 chocolate candies are wrapped. The calculation shows these events are dependent—knowing a candy is chocolate changes the probability it's wrapped. - At a school carnival, a student randomly selects one ticket from a box containing 40 tickets: 18 are blue, 12 are red, and 10 are green. Some tickets are marked as "prize" tickets: 14 of the 40 tickets are prize tickets, including 6 blue prize tickets, 5 red prize tickets, and 3 green prize tickets. Let Event A be "the ticket is blue" and Event B be "the ticket is a prize ticket." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(A∩B)=P(A)P(B).
This question tests understanding of independent events versus mutually exclusive events. To check if events A and B are independent, we need to verify if P(A∩B) = P(A)·P(B). From the given information: P(A) = 18/40, P(B) = 14/40, and P(A∩B) = 6/40 (blue prize tickets). Calculating: P(A)·P(B) = (18/40)·(14/40) = 252/1600 = 63/400, while P(A∩B) = 6/40 = 60/400. Since 63/400 ≠ 60/400, the events are not independent. Choice A incorrectly confuses independence with mutual exclusivity—the fact that some tickets are both blue AND prize tickets shows they're not mutually exclusive. - A quality inspector selects one item at random from a shipment. Let Event A be "the item is defective," and Event B be "the item is from Machine 1." The inspector is told that P(A∣B)>P(A). Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(A∣B)=P(A).
This question directly provides information about conditional probability. We're told P(A|B) > P(A), which means the probability of a defective item is higher when we know it's from Machine 1. This violates the independence condition P(A|B) = P(A). Choice B correctly identifies that the events are not independent because the conditional probability doesn't equal the unconditional probability. This makes intuitive sense - Machine 1 appears to produce defects at a higher rate than the overall average. Choice A incorrectly interprets the inequality. Choice D incorrectly applies the addition rule. Choice E incorrectly claims the events must be disjoint, but dependent events can still overlap. - A jar contains 6 red marbles and 4 blue marbles. Two marbles are drawn without replacement. Let Event A be "the first marble is red" and Event B be "the second marble is red." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because the first draw changes the composition of the jar for the second draw.
This question tests understanding of independence when sampling without replacement. Events A and B are not independent because drawing without replacement changes the probability distribution for the second draw. Initially, P(A) = 6/10 = 0.6. However, P(B|A) = 5/9 (only 5 red marbles remain out of 9 total) while P(B|A') = 6/9 (all 6 red marbles remain). Since P(B|A) ≠ P(B|A'), the events are dependent. The key insight is that without replacement, the first draw affects the composition of what remains, violating the independence condition P(A∩B) = P(A)·P(B).
- A student randomly selects one book from a shelf. The shelf has 6 fiction books and 4 nonfiction books; among the fiction books, 2 are mysteries. Let Event A be "the selected book is fiction" and Event B be "the selected book is a mystery." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because Event B is a subset of Event A.
This question probes independence when one event is a subset in book selection. Event A (fiction) has P(A) = 6/10 = 0.6, Event B (mystery) has P(B) = 2/10 = 0.2, and P(A ∩ B) = 0.2, but 0.6 * 0.2 = 0.12 ≠ 0.2, showing dependence due to B being subset of A. The relationship means selecting a mystery guarantees fiction, affecting probabilities. Distractor choice A wrongly assumes no overlap means exclusivity. Choice C misapplies union addition. Mini-lesson: Independence is violated when subset relations cause P(A ∩ B) ≠ P(A)P(B); mutual exclusivity requires zero overlap, distinguishing it from independence where overlap is possible but at the product rate.
- A bag contains 10 marbles: 4 red, 3 blue, and 3 yellow. Two marbles are drawn without replacement. Let Event A be "the first marble is red" and Event B be "the second marble is red." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because the outcome of the first draw changes the probability of the second draw.
This question examines independence in the context of sampling without replacement. When drawing without replacement, the first draw affects the probability of the second draw. Initially, P(B) = 4/10 (4 red marbles out of 10). However, if the first marble is red (Event A occurs), then P(B|A) = 3/9 (3 red marbles left out of 9 total). Since P(B|A) ≠ P(B), the events are not independent. The key insight is that without replacement creates dependence between successive draws. Choice E incorrectly suggests the events are mutually exclusive, but both events can occur (drawing two red marbles in a row is possible).
- In a survey, 48% of respondents prefer Brand X, 42% prefer Brand Y, and 10% prefer both (they said either would be fine). Let Event A be "a randomly selected respondent prefers Brand X" and Event B be "the respondent prefers Brand Y." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(A∩B) is not equal to P(A)P(B).
This question tests independence with overlapping preferences. Given P(A) = 0.48, P(B) = 0.42, and P(A∩B) = 0.10, we check if P(A∩B) = P(A)·P(B). We calculate P(A)·P(B) = 0.48 × 0.42 = 0.2016. Since 0.10 ≠ 0.2016, events A and B are not independent. The lower intersection probability (0.10 < 0.2016) indicates negative association - people tend to prefer one brand or the other, not both. This shows that events representing choices or preferences often exhibit dependence, as people's preferences tend to be exclusive rather than independent. - A spinner is divided into 8 equal sectors labeled 1 through 8. The spinner is spun once. Let Event A be "the result is a multiple of 2" and Event B be "the result is a multiple of 4." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(B∣A)=P(B).
This question examines independence when one event is a subset of another. Event A (multiples of 2) = {2, 4, 6, 8}, so P(A) = 4/8 = 1/2. Event B (multiples of 4) = {4, 8}, so P(B) = 2/8 = 1/4. Since every multiple of 4 is also a multiple of 2, B⊆A, meaning A∩B = B = {4, 8}, so P(A∩B) = 2/8 = 1/4. The conditional probability P(B|A) = P(A∩B)/P(A) = (1/4)/(1/2) = 1/2, while P(B) = 1/4. Since P(B|A) ≠ P(B), the events are not independent. When one event is a subset of another, they cannot be independent unless one has probability 0 or 1. - A school reports that 40% of students play a sport, 30% are in band, and 15% do both. Let Event A be "a randomly selected student plays a sport" and Event B be "the student is in band." Which statement about Events A and B is correct?
Answer: Events A and B are not independent because P(A∩B) is not equal to P(A)P(B).
This question tests the independence formula P(A∩B) = P(A)·P(B) with given probabilities. We have P(A) = 0.40, P(B) = 0.30, and P(A∩B) = 0.15. For independence, we need P(A∩B) = P(A)·P(B) = 0.40 × 0.30 = 0.12. Since 0.15 ≠ 0.12, events A and B are not independent. The fact that P(A∩B) > P(A)·P(B) indicates positive association - students who play sports are more likely than average to be in band. This demonstrates that independence requires the exact equality P(A∩B) = P(A)·P(B), not just that events can occur together. - A website tracks whether a visitor is on a mobile device and whether they make a purchase. Let Event A be "visitor is on mobile," and Event B be "visitor makes a purchase." The site reports that P(B∣A)=P(B). Which statement about Events A and B is correct?
Answer: Events A and B are independent because P(B∣A)=P(B).
This question tests understanding of conditional probability and independence. The given condition P(B|A) = P(B) is exactly the definition of independence - it means that knowing a visitor is on mobile doesn't change the probability they make a purchase. This is equivalent to saying P(A∩B) = P(A)P(B), which is the multiplication rule for independent events. Choice A correctly identifies this relationship. Choices C and D give incorrect formulas for independence. Choice B incorrectly suggests the events must be disjoint, but independent events can (and often do) occur together. Choice E incorrectly assumes temporal order affects independence. - A school cafeteria sells 3 types of fruit cups: apple, berry, and tropical. One fruit cup is selected at random from a large bin. Let Event A be "the cup is berry," and Event B be "the cup is labeled 'low sugar.'" Some berry cups are low sugar, and some non-berry cups are also low sugar. Which statement about Events A and B is correct?
Answer: Events A and B are independent if and only if P(A∩B)=P(A)P(B).
This question tests understanding of the definition of independent events. The correct answer is B because independence is defined by the multiplication rule: P(A∩B) = P(A)P(B). The problem states that some berry cups are low sugar, meaning events A and B can occur together, so they are not mutually exclusive. Choice A incorrectly confuses mutual exclusivity with independence - events that cannot occur together are mutually exclusive, not independent. Choice C gives the formula for the union of mutually exclusive events, not independent events. Independence means that knowing one event occurred doesn't change the probability of the other event, which is captured by the multiplication rule. - A jar contains 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." Which statement about Events A and B is correct?
Answer: Events A and B are disjoint, and if both have positive probability, then they cannot be independent.
This question explores the relationship between disjoint (mutually exclusive) events and independence. Events A (red marble) and B (blue marble) cannot occur simultaneously since a marble cannot be both red and blue, making them disjoint with P(A∩B) = 0. For disjoint events with positive probabilities, P(A∩B) = 0 cannot equal P(A)P(B) > 0, so they cannot be independent. Choice B correctly identifies this principle. Choice A incorrectly claims P(A∩B) = 0 implies independence. Choice C correctly states the addition rule for disjoint events but doesn't address independence. The key insight is that disjoint events with positive probabilities are never independent because knowing one occurred tells you the other didn't. - A student randomly chooses one course from a list. Let Event A be "the course is an AP course," and Event B be "the course meets in the morning." The school counselor states: "P(A∪B) can be found by adding P(A) and P(B)." Which statement about Events A and B is correct?
Answer: The counselor is correct only if Events A and B are disjoint (mutually exclusive).
This question tests understanding of when the addition rule P(A∪B) = P(A) + P(B) applies. This formula is only valid when events A and B are disjoint (mutually exclusive), meaning they cannot occur together. For general events, we need P(A∪B) = P(A) + P(B) - P(A∩B). Choice B correctly identifies that the counselor's statement is only true if the events are disjoint. Since an AP course could meet in the morning, the events can overlap, so they may not be disjoint. Choice A incorrectly suggests independence is the requirement. Choice C confuses union with intersection. The distinction between disjoint events and independent events is crucial: disjoint events satisfy the simple addition rule, while independent events satisfy the multiplication rule. - A spinner is divided into 10 equal sections labeled 1 through 10. One spin is made. Let Event A be "the result is even" and Event B be "the result is greater than 6." Which statement about Events A and B is correct?
Answer: Events A and B are independent because P(A∩B)=P(A)P(B).
This question evaluates independence in a spinner outcome by testing P(A ∩ B) = P(A)P(B). Event A (even) has P(A) = 5/10 = 0.5, Event B (>6) has P(B) = 4/10 = 0.4, and P(A ∩ B) = 2/10 = 0.2, equaling 0.5 * 0.4 = 0.2, confirming independence. The relationship shows evenness and being >6 are unrelated on this spinner. Distractor choice B wrongly assumes no overlap means exclusivity without checking. Choice C misuses the union formula for independence. Mini-lesson: Independent events satisfy the product rule for intersection, while mutually exclusive events have zero intersection; the distinction is crucial, as independent events can overlap, but exclusive ones cannot and are thus dependent.
