Question

If $P\left(\frac{A}{B}\right)=0.2,P\left(B\right)=0.5$ and P(A)=0.2, then $10P(A\cap \overline{B})$ = ___

Answer 1

Let us consider the venn diagram of A and B to find $P(A\cap \overline{B})$

The shaded region is P(A). It can also be obtained by removing A intersection B from A
$\Rightarrow P(A\cap \overline{B})=P\left(A\right)-P(A\cap B)$
We are given $P\left(\frac{A}{B}\right)=0.2$
If we know $P(A\cap B)$, then we can find the value of $P(A\cap \overline{B})$
For that we can use the expression for finding $P\left(\frac{A}{B}\right)$. We use this because it also has $P(A\cap B)$ and other variables are known to us.
$P\left(\frac{A}{B}\right)=\frac{P(A\cap B)}{P\left(B\right)}$
$\Rightarrow P(A\cap B)=P\left(\frac{A}{B}\right)P\left(B\right)$
$=0.5\times 0.2$
$=0.1$
$\Rightarrow P(A\cap \overline{B})=P\left(A\right)-P(A\cap B)$
$=0.2-0.1$
$=0.1$
$\Rightarrow 10P(A\cap \overline{B})=1$