Question 1119128: A fair die is tossed. Consider events A = {2, 4, 6}, B = {1, 2}, C = {1, 2, 3, 4}. Find:
1. P(A and B) and P(A or C)
2. P(A|B) and P(B|A)
3. P(A|C) and P(C|A)
4. P(B|C) and P(C|B)
Answer by greenestamps(13198)   (Show Source):
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Given A = {2, 4, 6}, B = {1, 2}, C = {1, 2, 3, 4}, we can find
(A and B) = {2} (all the elements in BOTH A AND B); (A or C) = {1, 2, 3, 4, 6}. (all the elements in EITHER A OR C)
Question (1): So then P(A and B) = 1/6; P(A or C) = 5/6.
For the conditional probability problems like P(A|B), I find it easiest to view the problem as B being the sample space, and the "good" elements are the elements of B that are also elements of A. So...
P(A|B): B contains two elements, 1 and 2. Of those, one (2) is also in A. So P(A|B) is 1/2.
P(B|A): A contains three elements, 2, 4, and 6. Of those, one (2) is also in B. So P(B|A) is 1/3.
You can answer the others in a similar manner. Here are the types of questions you need to ask:
P(A|C): What fraction of the elements of C are also elements of A?
P(C|A): What fraction of the elements of A are also elements of C?
and likewise for P(B|C) and P(C|B).
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