The probability that a man and his wife live after \(20\) years are \(\dfrac{1}{4}\) and \(\dfrac{1}{3}\) respectively. The probability that neither the man nor his wife live after \(20\) years is

Let \(P(M)\) be the probability that the man lives after \(20\) years and \(P(W)\) be the probability that his wife lives after \(20\) years.

Given \(P(M) = \dfrac{1}{4}\) and \(P(W) = \dfrac{1}{3}\).

The probability that the man does not live after \(20\) years is \(P(\overline{M}) = 1 - P(M) = 1 - \dfrac{1}{4} = \dfrac{3}{4}\).

The probability that the wife does not live after \(20\) years is \(P(\overline{W}) = 1 - P(W) = 1 - \dfrac{1}{3} = \dfrac{2}{3}\).

Assuming the events are independent, the probability that neither of them lives after \(20\) years is \(P(\overline{M} \cap \overline{W}) = P(\overline{M}) \times P(\overline{W})\).

\(P(\overline{M} \cap \overline{W}) = \dfrac{3}{4} \times \dfrac{2}{3} = \dfrac{1}{2}\).

Answer: \(\dfrac{1}{2}\)