Independent survival over 10 years. If P(man alive after 10 years) = 1/4 and P(wife alive after 10 years) = 1/3, what is P(neither alive after 10 years)?

Introduction / Context:
Assuming independence of survival between spouses, the probability that neither is alive equals the product of their individual death probabilities.

Given Data / Assumptions:

• P(man alive) = 1/4 ⇒ P(man dies) = 3/4.
• P(wife alive) = 1/3 ⇒ P(wife dies) = 2/3.
• Independence of outcomes.

Concept / Approach:
P(neither alive) = P(man dies) × P(wife dies) under independence.

Step-by-Step Solution:
P(neither) = (3/4) × (2/3) = 1/2.

Verification / Alternative check:
Complement: P(at least one alive) = 1 − 1/2 = 1/2. This equals P(man alive) + P(wife alive) − P(both alive) = 1/4 + 1/3 − (1/4 × 1/3) = 7/12 − 1/12 = 1/2.

Why Other Options Are Wrong:
1/12 is the probability both are alive; 3/4 and 7/12 are not the required joint death probability.

Common Pitfalls:
Confusing survival with death probabilities or assuming dependence without evidence.

Final Answer:
1/2