Question

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

• A. $\frac{3}{4}$
• B. $\frac{5}{12}$
• C. $\frac{7}{12}$
• D. $\frac{1}{2}$

Hint:

"Neither A nor B" translates to $(A' \cap B')$. For independent events, always calculate the individual complementary probabilities first ($1-p$), and then multiply them together. Don't confuse it with $1 - P(A \cap B)$.

Answer 1

The Correct Option is D

To find the probability that neither the man nor his wife lives after 20 years, we need to first understand the given probabilities and then use the concept of complementary probability.

1. The probability that the man is alive after 20 years is given as \(\frac{1}{4}\). Therefore, the probability that the man is not alive after 20 years is: \(1 - \frac{1}{4} = \frac{3}{4}\).
2. The probability that the wife is alive after 20 years is given as \(\frac{1}{3}\). Therefore, the probability that the wife is not alive after 20 years is: \(1 - \frac{1}{3} = \frac{2}{3}\).
3. Since the events of the man and wife surviving are independent, the probability that neither the man nor his wife is alive after 20 years is the product of their individual probabilities of not being alive: \( \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2} \).

Thus, the probability that neither the man nor his wife is alive after 20 years is \(\frac{1}{2}\).

This matches with the correct answer given.