Question

We are given: ?(? = 0) = .4 and ?(? = 1) = .6 . And: ?(? = 0|? = 0) = .95 and ?(? = 1|? = 1) = .95 ?(? = 1|? = 0) = .05 and ?(? = 0|? = 1) = .05

We must find P(B = 0 | A = 0).

Answer 1

We are given here that:
P(B = 0) = 0.4,
P(B = 1) = 0.6

Also, we are given that:
P(A = 0 | B = 0) = 0.95, and P(A = 1 | B = 0) = 0.05
P(A = 1 | B = 1) = 0.95, and P(A = 0 | B = 1) = 0.05

Using law of total probability, we get here:
P(A = 0) = P(A = 0 | B = 0) P(B = 0) + P(A = 0 | B = 1) P(B = 1)

P(A = 0) = 0.95*0.4 + 0.05*0.6 = 0.41

Using Bayes theorem, we get here:
P(B = 0 | A = 0) = P(A = 0 | B = 0) P(B = 0) / P(A = 0) = 0.95*0.4 / 0.41 = 0.9268

Therefore 0.9268 is the required probability here.