The probabilities that A and B will die with in a year are p and q respectively, then the probability that only one of them will be alive at the end of the year is -

Options

A. P + q
B. P + q - pq
C. p + q + pq
D. p + q - 2pq

Answer

To solve the problem, we need to find the probability that only one of A or B will be alive at the end of the year, given that the probabilities of A and B dying within a year are \( p \) and \( q \) respectively.

### Step-by-Step Solution:

1. **Identify the Probabilities**:
- The probability that A dies within a year is \( P(A) = p \).
- The probability that B dies within a year is \( P(B) = q \).
- The probability that A survives (does not die) is \( P(A') = 1 - p \).
- The probability that B survives (does not die) is \( P(B') = 1 - q \).

2. **Define the Events**:
- We want to find the probability that only one of them is alive at the end of the year. This can happen in two scenarios:
- Scenario 1: A dies and B survives.
- Scenario 2: A survives and B dies.

3. **Calculate the Probability for Each Scenario**:
- For Scenario 1 (A dies and B survives):
\[ P(A \text{ dies and } B \text{ survives}) = P(A) \times P(B') = p \times (1 - q) \]
- For Scenario 2 (A survives and B dies):
\[ P(A' \text{ survives and } B \text{ dies}) = P(A') \times P(B) = (1 - p) \times q \]

4. **Combine the Probabilities**:
- Since these two scenarios are mutually exclusive, we can add their probabilities:
\[ P(\text{only one alive}) = P(A \text{ dies and } B \text{ survives}) + P(A' \text{ survives and } B \text{ dies}) \]
\[ = p(1 - q) + (1 - p)q \]

5. **Simplify the Expression**:
- Expanding the expression:
\[ = p - pq + q - pq \]
\[ = p + q - 2pq \]

6. **Final Result**:
- Therefore, the probability that only one of them will be alive at the end of the year is:
\[ P(\text{only one alive}) = p + q - 2pq \]

### Answer:
The probability that only one of them will be alive at the end of the year is \( p + q - 2pq \).