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Question

Event A occurs with probability 0.4. The conditional probability that A occurs given that B occurs is 0.5, while the conditional probability that A occurs given that B does not occur is 0.2. What is the conditional probability that B occurs given that A occurs?

Answer 1

GIVEN: P(A) = 0.4 ; P(A | B) = 0.5 ; P(A | B') = 0.2

FIND: P(B | A)

By the Law of Total Probability, P(A) = P(A | B) * P(B) + P(A | B') * P(B'). Since P(B') = 1 - P(B), we have:

P(A) = 0.4 = P(A | B) * P(B) + P(A | B') * [1 - P(B)]

Hence, by substitution:

0.4 = 0.5 * P(B) + 0.2 * [ 1 - P(B)] -->

0.4 = 0.5 * P(B) + 0.2 - 0.2 * P(B) -->

0.3 * P(B) = 0.2 -->

P(B) = 2/3 = 0.667. Now:

By definition, P(AB) = P(A | B) * P(B) -->

P(AB) = 0.5 * (2/3) = 1/3 = 0.333

So:

P(B | A) = P(BA) / P(A) = P(AB) / P(A) = (1/3) / 0.4 -->

P(B | A) = 5 / 6 = 0.8333 (ANSWER)

Answer 2

A occurs 4 out of 10 times
B occurs 5 out of 10 times or 1 in 2 occasions ie odds

A will occur with equal odds on whether B occurs or not. Therefor 0.2.

Answer 3

HUH?