Independent events and filling in the Venn diagram

Text

Two events $A$ and $B$ are independent, and $P(A)=0.8$ and $P(B)=0.4$. Draw the Venn diagram, filling in all regions with their probabilities. (figure in the original test)

Solution

By independence $P(A\cap B)=P(A)\,P(B)=0.8\cdot0.4=0.32$. The regions of the diagram are then:

• $A$ only: $P(A)-P(A\cap B)=0.8-0.32=0.48$;
• $A\cap B$: $0.32$;
• $B$ only: $P(B)-P(A\cap B)=0.4-0.32=0.08$;
• outside $P(\bar A\cap\bar B)=1-(0.48+0.32+0.08)=0.12$. $\boxed{\ A\text{ only}=0.48;\ \ A\cap B=0.32;\ \ B\text{ only}=0.08;\ \ \text{outside}=0.12\ }$