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