Question

If $P\left(A\right)=\frac{2}{3}$, $P\left(B\right)=\frac{1}{2}$, and $P\left(AUB\right)=\frac{5}{6}$, then events $A$ and $B$ are

• A. Mutually exclusive
• B. Independent as well as mutually exhaustive
• C. Independent
• D. Dependent only on A

Answer 1

The correct option is C

Independent

Explanation for the correct option.

Step 1: Given inforamtion

Independent events: If two events are independent then $P(A\cap B)=P\left(A\right).P\left(B\right)$.

Given, $P\left(A\right)=\frac{2}{3}$, $P\left(B\right)=\frac{1}{2}$, $P\left(AUB\right)=\frac{5}{6}$.

Step 1: Find the value of $P(A\cap B)$.

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

Substitute the values we get

$\Rightarrow \frac{5}{6}=\frac{2}{3}+\frac{1}{2}–P(A\cap B)$

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

Step 2: Find the value of $P\left(A\right)·P\left(B\right)$.

$\begin{array}{rcl}\therefore P\left(A\right)·P\left(B\right)& =& \frac{2}{3}·\frac{1}{2}\\ & =& \frac{1}{3}\end{array}$

$\Rightarrow P(A\cap B)=P\left(A\right).P\left(B\right)$, therefore events are independent.

Hence, option C is correct.