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.

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Solution

## $P\left(\text{A solving the problem}\right)=P\left(A\right)=\frac{2}{3}\phantom{\rule{0ex}{0ex}}P\left(\text{B solving the problem}\right)=P\left(B\right)=\frac{3}{5}\phantom{\rule{0ex}{0ex}}\text{We need to find out if}P\left(\overline{A}\cap \overline{B}\right)=P\left(\overline{A}\right)P\left(\overline{B}\right)\left[\text{A and B are independent events}\right]\phantom{\rule{0ex}{0ex}}=\left[1-P\left(A\right)\right]\left[1-P\left(B\right)\right]\phantom{\rule{0ex}{0ex}}=\left(1-\frac{2}{3}\right)\left(1-\frac{3}{5}\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}×\frac{2}{5}\phantom{\rule{0ex}{0ex}}=\frac{2}{15}\phantom{\rule{0ex}{0ex}}$

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