Chapter 4.5, Problem 39E

The prior probabilities for events $A_1$ and $A_2$ are $P\left(A_1\right)=.40$ and $P\left(A_2\right)=.60$ .It is also known that $P\left(A_1\cap A_2\right)=0$ . suppose $P\left(B|A_1\right)=.20$ and $P\left(B|A_2\right)=\mathrm{.05.}$

1. Are $A_1$ and $A_2$mutually exclusive? Explain.

Compute $P\left(A_1\cap B\right)$ and $P\left(A_2\cap B\right).$

Compute $P\left(B\right).$

Apply Bayes' theorem to compute $P\left(A_1|B\right)$ and $P\left(A_2|B\right)$ .