A couple has two childen . find the probability that both are boys ., if it is known that the older child is a boy .

Options

A. `3/8`
B. `1/2`
C. `5/8`
D. `3/4`

Answer

To solve the problem, we need to find the probability that both children are boys given that the older child is a boy.

### Step-by-Step Solution:

1. **Identify the Sample Space**:
The possible combinations of two children (where B = boy and G = girl) are:
- BB (both are boys)
- BG (older is a boy, younger is a girl)
- GB (older is a girl, younger is a boy)
- GG (both are girls)

However, since we know that the older child is a boy, we can eliminate the combinations where the older child is a girl. Thus, the relevant sample space is:
- BB
- BG

2. **Define Events**:
- Let event A be the event that both children are boys (BB).
- Let event B be the event that the older child is a boy.

3. **Count the Outcomes**:
- The outcomes in event A (both are boys) is: {BB}
- The outcomes in event B (older is a boy) is: {BB, BG}

4. **Find the Intersection of Events A and B**:
- The intersection of A and B (A ∩ B) is the outcome where both children are boys and the older child is a boy, which is: {BB}
- Thus, the number of outcomes in A ∩ B is 1.

5. **Calculate the Probability**:
- The probability of A given B (P(A|B)) can be calculated using the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
- We know:
- \( P(A \cap B) = \text{Number of outcomes in A ∩ B} = 1 \)
- \( P(B) = \text{Number of outcomes in B} = 2 \) (which are BB and BG)

Therefore:
\[ P(A|B) = \frac{1}{2} \]

6. **Conclusion**:
The probability that both children are boys given that the older child is a boy is \( \frac{1}{2} \).