Question

If $P\left(A\right)=\frac{1}{4},P\left(B\right)=\frac{2}{5}$ and $P(A\cup B)=\frac{1}{2}$, find the value of $P(A\cap B)$

Answer 1

We have,

$P\left(A\right)=\frac{1}{4}$

$P\left(B\right)=\frac{2}{5}$

$P(A\cup B)=\frac{1}{2}$

So,

$P(A\cap B)=?$

Now, using formula,

$P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$

$P(A\cap B)=\frac{1}{4}+\frac{2}{5}-\frac{1}{2}$

$P(A\cap B)=\frac{2}{5}-\frac{1}{4}$

$P(A\cap B)=\frac{8-5}{20}$

$P(A\cap B)=\frac{3}{20}$

Hence, this is the answer.