Question

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

• A. dependent
• B. independent
• C. mutually exclusive
• D. none of these

Answer 1

The correct option is B independent

Given,

$P\left(B^{\prime }\right)=\frac{1}{2}$

or, $P\left(B\right)=1-\frac{1}{2}$

or $P\left(B\right)=\frac{1}{2}$.......(1).

Again given,

$P(A\cap B)=\frac{1}{3}$.......(2).

Also,

$P(A\cup B)=\frac{5}{6}$

or $P\left(A\right)+P\left(B\right)-P(A\cap B)=\frac{5}{6}$ [ Using formula]

or, $P\left(A\right)+\frac{1}{2}-\frac{1}{3}=\frac{5}{6}$ [ Using (1) and (2)]

or, $P\left(A\right)=\frac{2}{3}$.

Now we have $P(A\cap B)=P\left(A\right).P\left(B\right)$ [ Since $\frac{1}{3}=\frac{2}{3}.\frac{1}{2}$]

So the events $A,B$ are independent.