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

Step 1: Understanding the Concept:
The problem involves the probabilities of survival for two individuals. In such problems, it is standard to assume that their lifespans are independent events. We need to find the probability of the intersection of their respective complement events (neither lives).

Step 2: Key Formula or Approach:
1. Let $A$ be the event the man lives and $B$ be the event the wife lives. Identify $P(A)$ and $P(B)$. 2. The event "neither lives" means "the man does not live AND the wife does not live". This is the intersection of the complement events: $P(A' \cap B')$. 3. Calculate the complement probabilities: $P(A') = 1 - P(A)$ and $P(B') = 1 - P(B)$. 4. Use the multiplication rule for independent events: $P(A' \cap B') = P(A') \times P(B')$.

Step 3: Detailed Explanation:
Let $M$ be the event that the man lives after 20 years. We are given $P(M) = \frac{1}{4}$. The probability that the man does not live after 20 years is $P(M')$. \[ P(M') = 1 - P(M) = 1 - \frac{1}{4} = \frac{3}{4} \] Let $W$ be the event that the wife lives after 20 years. We are given $P(W) = \frac{1}{3}$. The probability that the wife does not live after 20 years is $P(W')$. \[ P(W') = 1 - P(W) = 1 - \frac{1}{3} = \frac{2}{3} \] We are asked to find the probability that neither lives after 20 years. This means the man dies AND the wife dies. Assuming their survival probabilities are independent: \[ P(\text{neither lives}) = P(M' \text{ and } W') = P(M' \cap W') \] \[ P(M' \cap W') = P(M') \times P(W') \] Substitute the calculated values: \[ P(\text{neither lives}) = \left(\frac{3}{4}\right) \times \left(\frac{2}{3}\right) \] \[ P(\text{neither lives}) = \frac{3 \times 2}{4 \times 3} \] \[ P(\text{neither lives}) = \frac{6}{12} \] Simplifying the fraction: \[ P(\text{neither lives}) = \frac{1}{2} \]

Step 4: Final Answer:
The probability that neither the man nor his wife live after 20 years is $\frac{1}{2}$.