Independent events from conditional probabilities

Text

Two events $A$ and $B$ are independent, and $P(A|B)=0.4$ and $P(B|A)=0.6$. Draw the Venn diagram, filling in all regions with their probabilities. (figure in the original test)

Solution

For independent events conditioning does not change the probabilities: $P(A|B)=P(A)=0.4$ and $P(B|A)=P(B)=0.6$. So $P(A\cap B)=P(A)\,P(B)=0.4\cdot0.6=0.24$. The regions of the diagram are:

• $A$ only: $0.4-0.24=0.16$;
• $A\cap B$: $0.24$;
• $B$ only: $0.6-0.24=0.36$;
• outside: $1-(0.16+0.24+0.36)=0.24$. $\boxed{\ P(A)=0.4;\ P(B)=0.6;\ A\cap B=0.24;\ A\text{ only}=0.16;\ B\text{ only}=0.36;\ \text{outside}=0.24\ }$