Question

Suppose events A, B, C have the following probabilities: P(A|B) = 1 ／3 , P(C|B) = 23 ／45 , P(A ∩ C|B) = 11 ／45 . Given that B has occurred, a) Find the probability that only C has occurred. b) Find the probability that only A or only C has occurred, but not both. c) Find the probability that A or C has occurred.

Answer 1

P(A n C | B) = 11/45 => A and C both occur given B occurs

P(A | B) = A occurs, given B occurs = 1/3

P(A | B) = P(A occurs and C occurs | B) + P(A occurs but C doesn't occur | B)

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FINAL ANSWERS

=> P(A occurs but C doesn't occur | B) = 1/3 - 11/45 = 4/45 [Answer b-Part 1]

Similarly,

P(C occurs but A doesn't occur | B) = 23/45 - 11/45 = 12/45 [Answer b-Part 2] and [Answer Part a]

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P(only A or Only C has occured | B) = 16/45 [Answer Part b]

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P(A or C have occured | B) = P(A | B) + P(C | B) - P(A n C | B) = 27/45 [Answer c]