Question

Suppose that antonia rolls a pair of fair six-sided dice. let \[a\] be the event that the first die lands on \[5\] and \[b\] be the event that the sum of the two dice is \[6\]. Using the sample space of possible outcomes below, answer each of the following questions. what is \[p(a)\], the probability that the first die lands on \[5\]? What is \[p(b)\], the probability that the sum of the two dice is \[6\]? what is \[p(a\text{ and }b)\], the probability that the first die lands on \[5\] and the two dice add up to \[6\]? are events \[a\] and \[b\] independent? Choose 1 answer: choose 1 answer: (choice a) yes, events \[a\] and \[b\] are independent events. a yes, events \[a\] and \[b\] are independent events. (choice b) no, events \[a\] and \[b\] are not independent events. b no, events \[a\] and \[b\] are not independent events.

Answer 1

The probability of the first die landing on a 5 (event A) is 1/6, the probability of the sum of two dice being 6 (event B) is 5/36, and the probability of both A and B occurring together (event A and B) is 1/36. Since P(A and B) does not equal P(A) × P(B), events A and B are not independent.

Step-by-step explanation:

When Antonia rolls a pair of fair six-sided dice, we can calculate probability for different events as outlined in the question.

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• The probability that the first die lands on 5, P(A), is 1/6 since there is one favorable outcome (landing a 5) out of six possible outcomes on a single die.
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• The probability that the sum of the two dice is 6, P(B), can be found by considering all the pairs that add up to 6: (1,5), (2,4), (3,3), (4,2), and (5,1). There are five favorable outcomes out of the 36 possible outcomes when rolling two dice, so P(B) is 5/36.
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• To find P(A and B), the probability that the first die lands on 5 and the sum of the two dice is 6, we look for the intersection of events A and B. The only pair that satisfies both conditions is (5,1), which gives us P(A and B) = 1/36.

Events A and B are independent if the occurrence of one does not affect the probability of occurrence of the other. We can use the definition of independence, which says events A and B are independent if P(A and B) = P(A) × P(B). Here, P(A and B) = 1/36, while P(A) × P(B) is (1/6) × (5/36) = 5/216, which is not equal to 1/36. Therefore, events A and B are not independent.