Understanding Probability of Lifespan Events

This question asks us to find the probability that neither a man nor his wife will live for another 10 years, given their individual probabilities of living for that duration.

Let's define the events:

• Let M be the event that the man lives for 10 more years.
• Let W be the event that his wife lives for 10 more years.

We are given the following probabilities:

• Probability that the man will live for 10 more years, $P(M) = \frac{1}{4}$.
• Probability that his wife will live for 10 more years, $P(W) = \frac{1}{3}$.

We need to find the probability that none of them will live for 10 more years. This means the man does not live for 10 more years AND the wife does not live for 10 more years.

Let M' be the event that the man does not live for 10 more years, and W' be the event that the wife does not live for 10 more years.

The probability of a complementary event (an event not happening) is 1 minus the probability of the event happening. So:

• Probability that the man will not live for 10 more years, $P(M') = 1 - P(M) = 1 - \frac{1}{4}$.

Calculation for $P(M')$:

\( P(M') = 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{4-1}{4} = \frac{3}{4} \)

• Probability that his wife will not live for 10 more years, $P(W') = 1 - P(W) = 1 - \frac{1}{3}$.

Calculation for $P(W')$:

\( P(W') = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{3-1}{3} = \frac{2}{3} \)

We want to find the probability that none of them will live for 10 more years, which is the probability of event M' and event W' occurring together. We assume that the man's lifespan is independent of the wife's lifespan. For independent events, the probability that both events occur is the product of their individual probabilities.

Probability that none of them will live for 10 more years is $P(M' \text{ and } W') = P(M') \times P(W')$.

Calculation for $P(M' \text{ and } W')$:

\( P(M' \text{ and } W') = \frac{3}{4} \times \frac{2}{3} \)

Multiplying the fractions:

\( \frac{3}{4} \times \frac{2}{3} = \frac{3 \times 2}{4 \times 3} = \frac{6}{12} \)

Simplifying the fraction:

\( \frac{6}{12} = \frac{1}{2} \)

Thus, the probability that none of them will live for 10 more years is $\frac{1}{2}$.

Let's check this result against the given options.

The calculated probability of $\frac{1}{2}$ matches option (c).

The final answer is $\frac{1}{2}$.

Revision Table: Probability Concepts

Additional Information: Independent vs. Dependent Probability

In probability, understanding whether events are independent or dependent is crucial.

Independent Events:

• Two events are independent if the outcome of one event does not influence the outcome of the other.
• For example, flipping a coin twice - the result of the first flip does not affect the second flip.
• The probability of both independent events A and B occurring is $P(A \text{ and } B) = P(A) \times P(B)$.

Dependent Events:

• Two events are dependent if the outcome of one event affects the outcome of the other.
• For example, drawing two cards from a deck without replacement - the probability of drawing a certain card on the second draw depends on what was drawn first.
• The probability of both dependent events A and B occurring is $P(A \text{ and } B) = P(A) \times P(B|A)$, where $P(B|A)$ is the conditional probability of B occurring given that A has already occurred.

In this problem, it is reasonable to assume the lifespans of a husband and wife are independent events in this context, although in reality, factors might influence both (like shared lifestyle). However, for typical probability problems of this type, independence is assumed unless stated otherwise.