How to find $\Pr(A)$ given $\Pr(B \mid A)$, $\Pr(B \mid \neg A)$ and $\Pr(B)$

Question

The conditional probability $\Pr[B \mid A]$ is $\frac{4}{5}$; the conditional probability $\Pr[B \mid \neg A]$ is $\frac{2}{5}$, and the unconditional probability of $B$ is $\frac{1}{2}$. What is the probability of $A$?

Answer 1

By the law of total probability, we have

$$\Pr(B) = \Pr(B\mid A)\Pr(A) + \Pr(B\mid \lnot A)\Pr(\lnot A)$$

Now, you know everything in the above equation except for $\Pr(A)$, so you can easily solve for the answer.