Question

If $P\left(A\right)=\frac{2}{3}$, $P\left(B\right)=\frac{2}{5}$ and $P(A\cup B)-P(A\cap B)=\frac{2}{5}$, then what is $P(A\cap B)$ equal to?

• A. $\frac{3}{5}$
• B. $\frac{5}{11}$
• C. $\frac{1}{3}$
• D. None of the above

Answer 1

Answer: (C)

$\frac{1}{3}$

The sets rule is $P\left(A\right)+P\left(B\right)=P(A\cup B)+P(A\cap B)$

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

$P\left(A\right)+P\left(B\right)=\frac{2}{3}+\frac{2}{5}=\frac{16}{15}$

So $P(A\cup B)+P(A\cap B)=\frac{16}{15}$

And given $P(A\cup B)-P(A\cap B)=\frac{2}{5}$

So $P(A\cup B)=\frac{\frac{16}{15}+\frac{2}{5}}{2}=\frac{22}{30}=\frac{11}{15}$

$\Rightarrow P(A\cup B)=\frac{11}{15}\therefore P(A\cap B)=\frac{11}{15}-\frac{2}{5}=\frac{1}{3}$