Question

The probability for a contractor to get a road contract is $\frac{2}{3}$ and to get a building contract is $\frac{5}{9}$. The probability to get at least one contract is $\frac{4}{5}$. Find the probability that he gets both the contracts

Answer 1

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

Using the above relation, we can write

$\frac{4}{5}=\frac{2}{3}+\frac{5}{9}-P(A\cap B)$

$\therefore P(A\cap B)=\frac{2}{3}+\frac{5}{9}-\frac{4}{5}$

$=\frac{45\times 2+5\times 15-4\times 27}{135}$

$=\frac{90+75-108}{135}$

$=\frac{57}{135}$

$\therefore P(A\cap B)=\frac{19}{45}$

The probability of him getting both the contracts is $\frac{19}{45}$