Question

If A and B are two events such that $P(A\cup B)=\frac{5}{6},P(A\cap B)=\frac{1}{3},P\left(\overline{B}\right)=\frac{1}{2}$, then the events A and B are

• A. dependent
• B. independent
• C. mutually exclusive
• D. None of the above

Answer 1

The correct option is B independent

$P(A\cup B)=\frac{5}{6},P(A\cap B)=\frac{1}{3},P\stackrel{\_}{(B})=\frac{1}{2}\Rightarrow P(B)=1-\frac{1}{2}=\frac{1}{2}P(A\cup B)=P(A)+P(B)-P(A\cap B)\frac{5}{6}=P(A)+\frac{1}{2}-\frac{1}{3}\Rightarrow P(A)=\frac{2}{3}P(A).P(B)=\frac{2}{3}.\frac{1}{2}=\frac{1}{3}\Rightarrow P(A\cap B)=P(A).P(B)$

So the two events are independent events

Option $B$ is correct.