Please show work not just final answers!! Thanks. 86. Roll two fair dice. Each d
ID: 3440082 • Letter: P
Question
Please show work not just final answers!! Thanks.
86. Roll two fair dice. Each die has six faces. a. List the sample space. b. Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A). C. Let B be the event that the sum of the two rolls is at most seven. Find P(B). d. In words, explain what P(A|B) represents. Find P(A|B). e. Are A and B mutually exclusive events? Explain your answer in one to three complete sentences, including numerical justification. f. Are A and B independent events? Explain your answer in one to three complete sentences, including numerical justification.Explanation / Answer
a)
These are the possible results:
b)
Counting these events, there are 6 such events.
hence,
P(3 or 4, then even) = 6/36 = 1/6 [answer]
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c)
Consider the table:
Here, there are 21 such events that the sum is at most 7. Hence,
P(at most 7) = 21/36 = 7/12 = 0.583333333 [answer]
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d)
P(A|B) means "the probability that a three or four occured first and it is followed by an even number, given that the sum is at most 7."
Among the 6 events of A, there are 3 of them with sum at most 7, namely, (3,2), (4,2), (3,4).
Thus,
P(B|A) = 3/6 = 0.5 [answer]
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e)
They are not, because they have intersection (can happen together).
As we can see, those events that they happened together are (3,2), (4,2), (3,4).
P(A and B) = 3/36 = 1/12 =/= 0.
Thus, they are not mutually exclusive.
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f)
They are not, as P(B}A) =/= P(B).
Die 1 Die 2 1 1 1 2 1 3 1 4 1 5 1 6 2 1 2 2 2 3 2 4 2 5 2 6 3 1 3 2 3 3 3 4 3 5 3 6 4 1 4 2 4 3 4 4 4 5 4 6 5 1 5 2 5 3 5 4 5 5 5 6 6 1 6 2 6 3 6 4 6 5 6 6