How to do Bayes' theorem?

Question

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places. P(A | B) = .9, P(B) = .6, P(A | B') = .8. Find P(B | A).

We don't have an example like this in class so I was wondering if someone could help me figure out how to solve it.

Answer 1

Bayes theorem:

$$P(B|A)=\frac{P(A|B)P(B)}{P(A)}$$

So you only need to find $P(A)$. Use the law of total probability:

$$P(A)=P(A|B)P(B)+P(A|B')P(B')$$

and notice that $$P(B')=1-P(B)$$