Question

Given the probability that A can solve a problem is 2/3 and the probability that B can solve the same problem is 3/5. Find the probability that none of the two will be able to solve the problem.

Answer 1

$$\begin{gathered} P\left(\text{A solving the problem}\right)=P\left(A\right)=\frac{2}{3} \\ P\left(\text{B solving the problem}\right)=P\left(B\right)=\frac{3}{5} \\ \text{We need to find out if}P\left(\overline{A}\cap \overline{B}\right)=P\left(\overline{A}\right)P\left(\overline{B}\right)\left[\text{A and B are independent events}\right] \\ =\left[1-P\left(A\right)\right]\left[1-P\left(B\right)\right] \\ =\left(1-\frac{2}{3}\right)\left(1-\frac{3}{5}\right) \\ =\frac{1}{3}\times \frac{2}{5} \\ =\frac{2}{15} \end{gathered}$$