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 ?

• A. $\frac{20}{115}$
• B. $\frac{17}{105}$
• C. $\frac{10}{343}$
• D. $\frac{9}{17}$

Answer 1

The correct option is B $\frac{17}{105}$

Consider the following events :

A = Person gets an electric contract, B = Person gets plumbing contract

We have,

$P\left(A\right)=\frac{2}{5},P\left(\overline{B}\right)=\frac{4}{7},P(A\cap B)=\frac{2}{3}$

Now,

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

$\frac{2}{3}=\frac{2}{5}+(1-\frac{4}{7})-P(A\cap B)$

$P(A\cap B)=\frac{2}{5}+\frac{3}{7}-\frac{2}{3}=\frac{17}{105}$