Roll two fair dice separately. Each dice has six faces. Let A be the event that either a three or four is rolled first, followed by an even number. Let B be the event that the sum of the two rolls is at most seven. Are A and B independent events?Question 2Answera.Yesb.No
Question
Roll two fair dice separately. Each dice has six faces. Let A be the event that either a three or four is rolled first, followed by an even number. Let B be the event that the sum of the two rolls is at most seven. Are A and B independent events?Question 2Answera.Yesb.No
Solution
To determine if events A and B are independent, we need to check if the probability of A happening does not affect the probability of B happening, and vice versa.
Event A: Either a three or four is rolled first, followed by an even number. The probability of rolling a three or four on a six-sided die is 2/6 or 1/3. The probability of rolling an even number (2, 4, or 6) is 3/6 or 1/2. So, the probability of event A is (1/3) * (1/2) = 1/6.
Event B: The sum of the two rolls is at most seven. There are 36 possible outcomes when rolling two dice (6 faces on the first die * 6 faces on the second die). The outcomes where the sum is at most seven are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (5,1), (5,2), (6,1). That's 21 outcomes, so the probability of event B is 21/36 or 7/12.
To check if A and B are independent, we need to see if P(A and B) = P(A) * P(B).
The outcomes where both A and B occur are: (3,2), (3,4), (4,2), (4,4). That's 4 outcomes, so P(A and B) = 4/36 = 1/9.
However, P(A) * P(B) = (1/6) * (7/12) = 7/72, which is not equal to 1/9.
Therefore, events A and B are not independent. So, the answer is b. No.
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