The probability a man survives 10 more years is 1/4 and the probability his wife survives 10 more years is 1/3. Assuming independence, what is the probability that neither of them is alive after 10 years?

Introduction / Context:Given two independent survival probabilities over a fixed horizon, we are asked for the probability that neither survives. This is a direct complement multiplication problem.

Given Data / Assumptions:

• P(man survives 10 yrs) = 1/4 → P(man does not survive) = 3/4.
• P(wife survives 10 yrs) = 1/3 → P(wife does not survive) = 2/3.
• Independence of the two events.

Concept / Approach:For independent events X and Y, P(X and Y) = P(X)*P(Y). Here X = “man does not survive,” Y = “wife does not survive.”

Step-by-Step Solution:P(neither survives) = (3/4) * (2/3) = 6/12 = 1/2.

Verification / Alternative check:Compute P(at least one survives) = 1 − 1/2 = 1/2. Alternatively, use inclusion–exclusion on survival events and complement.

Why Other Options Are Wrong:5/12 and 7/12 are near but result from incorrect sums/products; 11/12 is far too large.

Common Pitfalls:Adding the “non-survival” probabilities instead of multiplying for the joint event.

Final Answer:1/2