Conditional probability calculation

Question

Given $P(B \cap A') = 0.25$, $ P(A \cap B') = 0.2$ and $P(A' \mid B') = 0.5$, how do we find $P( A \cap B)$, and $P( B' \cap A')$?

(Here $A', B'$ are the complements of $A$ and $B$ respectively)

Answer 1

From venn diagrams

$P(only~ B)=0.25$, $P(only~A)=0.2$, $P(\overline{A\cup B})/P(\overline{B})=0.5$

Let $P[A\cap B]= x$

$$\implies \frac{0.55-x}{0.75-x}=0.5 \implies x=0.35$$

Now you can find the rest.