Question

If $P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$, and $P(A\cap B)=\frac{7}{12}$, then the value of $P\left(A^{\text{'}}\cap B^{\text{'}}\right)$ is

• A. $\frac{7}{12}$
• B. $\frac{3}{4}$
• C. $\frac{1}{4}$
• D. $\frac{1}{6}$

Answer 1

The correct option is B

$\frac{3}{4}$

Explanation for the correct option.

Step 1: Find the value of $P\left(A\cup B\right)$

Given, $P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$and $P(A\cap B)=\frac{7}{12}$.

Apply this formula:

$P\left(A^{\text{'}}\cap B^{\text{'}}\right)=P{\left(A\cup B\right)}^{\text{'}}$ and $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\therefore P\left(A^{\text{'}}\cap B^{\text{'}}\right)=1-P\left(A\cup B\right)$

$\begin{array}{rcl}\therefore P\left(A\cup B\right)& =& \frac{1}{2}+\frac{1}{3}-\frac{7}{12}\\ & =& \frac{6+4-7}{12}\\ & =& \frac{1}{4}\end{array}$

Step 2: Find the value of $P\left(A\text{'}\cap B\text{'}\right)$

$\begin{array}{rcl}\therefore P\left(A^{\text{'}}\cap B^{\text{'}}\right)& =& 1-\frac{1}{4}\\ & =& \frac{4-1}{4}\\ & =& \frac{3}{4}\end{array}$

Hence, option B is correct.