Question

If $P\left(A\right)=P\left(B\right)=x$and $P(A\cap B)=P(A\text{'}\cap B\text{'})=\frac{1}{3}$, then $x$ is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{6}$

Answer 1

The correct option is A

$\frac{1}{2}$

Explanation for the correct option.

Step 1: Determination of $P(A\cup B)$

Given that, $P\left(A\right)=P\left(B\right)=x$and $P(A\cap B)=P(A\text{'}\cap B\text{'})=\frac{1}{3}$.

Since the sum of the complement of the union of two sets and the union of two sets is equal to one then we have,

$P(A\text{'}\cap B\text{'})=P(A\cup B)\text{'}$ and $P(A\cup B)\text{'}=1-P(A\cup B)$.

From this, we can write that,

$$\begin{gathered} \therefore P(A\text{'}\cap B\text{'})=1-P(A\cup B) \\ \Rightarrow P(A\cup B)=1-\frac{1}{3} \\ \Rightarrow P(A\cup B)=\frac{2}{3} \end{gathered}$$

Step 2: Find the value of $x$

Now we know that $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$ we get,

$$\begin{gathered} \therefore \frac{2}{3}=x+x-\frac{1}{3} \\ \Rightarrow \frac{2}{3}+\frac{1}{3}=2x \\ \Rightarrow 1=2x \\ \Rightarrow x=\frac{1}{2} \end{gathered}$$

Hence, option (A) is correct.