Question

If two events $P\left(AUB\right)=\frac{5}{6},P(A\text{'})=\frac{5}{6},P\left(B\right)=\frac{2}{3}$then $A$ and $B$ are.

• A. Independent
• B. Mutually exclusive
• C. Mutually exhaustive
• D. Dependent

Answer 1

The correct option is B

Mutually exclusive

Explanation for the correct answer:

To find the required answer:

Given that $P\left(AUB\right)=\frac{5}{6}$

$P(A\text{'})=\frac{5}{6}$, $P\left(B\right)=\frac{2}{3}$

Then,

$\begin{array}{rcl}P\left(A\right)& =& 1-P\left(A\text{'}\right)\\ & =& \frac{1}{6}\end{array}$

We know, $P\left(AUB\right)=P\left(A\right)+P\left(B\right)–P(A\cap B)$

So,

$\begin{array}{rcl}\frac{5}{6}& =& \frac{1}{6}+\frac{2}{3}–P(A\cap B)\\ & \Rightarrow & \frac{5}{6}=\frac{(1+4)}{6}-P(A\cap B)\\ & \Rightarrow & \frac{5}{6}=\frac{5}{6}-P(A\cap B)\\ & \Rightarrow & P(A\cap B)=0\end{array}$

If $P(A\cap B)=0$ then two events are mutually exclusive.

Hence, the correct option is B.