Find $P(A^\complement \mid B)$ given the following information: $P(B \mid A)=0.4, P(A)=0.9, P(B \mid A^\complement)=0.7$...

Question

Here are the multiple-choice options:

(a) 0.163 (b) 0.25 (c) 0.423 (d) 0.512 (e) 0.724

I have tried a lot but only got 0.126, so I messed up somewhere along the way...

Answer 1

$$P(B|A) = 0.4, P(A) = 0.9, P(B|(1-A)) = 0.7$$

$$P((1-A)|B) = \dfrac{P(B|1-A)P(1-A)}{P(B)}$$

$$P(1-A) = 1-P(A) = 1-0.9 = 0.1$$

$$P(B) = P(B\cap A) + P(B\cap (1-A)) = P(B|A)P(A)+P(B|(1-A))P(1-A) = (0.4)(0.9)+(0.7)(0.1)$$

Putting it all together:

$$P((1-A)|B) = \dfrac{(0.7)(0.1)}{(0.4)(0.9)+(0.7)(0.1)} = \dfrac{7}{43} \approx 0.163$$