Question

The probability that a person will get an electric contract is $\frac{2}{5}$ and the probability that he will not get plumbing contract is $\frac{4}{7}$. If the probability of getting at least one contract is $\frac{2}{3}$, what is the probability that he will get both?

Answer 1

Let electricity contract be represented by E and plumbing contract be P

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

$P\left(P\right)=1-\frac{4}{7}=\frac{3}{7}$

$P\left(E\cup P\right)=\frac{2}{3}$

NOW $P\left(E\cap P\right)=P\left(E\right)+P\left(P\right)-P\left(E\cup P\right)$

$=\frac{2}{5}+\frac{3}{7}-\frac{2}{3}$

$=\frac{17}{105}$