Question

Let P(A) = 0.4, P(B|A) = 0.5, and P(B|A^c) = 0.25. Compute P(A|B) .

0.20

0.35

0.57

0.80

The total probability rule is defined as P(A) = P(A ∩ B) P(A∩ Bc)

True

False

Answer 1

Bayes' Theorem: P(A | B) = P(A B) / P(B)

P(A) = 0.4

P(A^c) = 1 - 0.4 = 0.6

P(B | A^c) = P(B A^c) / P(A^c)

0.25 = P(B A^c) / 0.6

P(B A^c) = 0.6x0.25 = 0.15

P(B | A) = P(B A) / P(A)

P(B A) = P(B | A) x P(A)

= 0.5 x 0.4

= 0.2

P(B A) + P(B A^c) = P(B)

P(A B) = P(B) - P(B A^c)

0.2 = P(B) - 0.15

P(B) = 0.35

P(A | B) = 0.2/0.35

= 0.57

Total probability rule: P(A) = P(A ∩ B) + P(A ∩ B^c)