If $P(A) ,\ P(B|A)$ and $P(B|A’)$ are given, how do we find the value of $P(B)$?

Question

If $P(A)=\frac{2}{5} , P(B|A)=\frac{1}{3} , P(B|A’)=\frac{1}{4}$ how do we find the value of $P(B)$?

I tried using the conditional probability formula but the question seems to be lacking information. I do believe that they are not independent events however

Answer 1

Either event $A$ happens or not, that is, $A'$ happens.

By definition, $P(B|A)=P(B\cap A)/P(A)$; similarly, $P(B|A')=P(B\cap A')/P(A')$.

So \begin{align*}P(B)&=P(B\cap A) + P(B\cap A')\\ &=P(B|A)P(A) + P(B|A')P(A')\\&=\frac{1}{3}\times\frac{2}{5}+\frac{1}{4}\times\frac{3}{5}=\frac{17}{60}\end{align*}