KCET 2026 Mathematics Solved Paper

Concept:When two events are independent, the probability of both happening is the product of their individual probabilities.Explanation:The probability that the man lives after 20 years is $P(M) = \frac{1}{4}$.The probability that his wife lives after 20 years is $P(W) = \frac{1}{3}$.So the probability that the man does not live after 20 years is$$P(\bar{M}) = 1 - P(M) = 1 - \frac{1}{4} = \frac{3}{4}.$$The probability that the wife does not live after 20 years is$$P(\bar{W}) = 1 - P(W) = 1 - \frac{1}{3} = \frac{2}{3}.$$Assuming the lifetimes of the man and his wife are independent, the probability that neither lives after 20 years is$$P(\bar{M} \cap \bar{W}) = P(\bar{M}) \times P(\bar{W}) = \frac{3}{4} \times \frac{2}{3} = \frac{1}{2}.$$Answer:The probability is $\frac{1}{2}$, which corresponds to option D.