Assume that in a family, each child is equally likely to be a boy or girls .A family with three children is is choosen at random. The probability that the eldest child is a girls given that the family has at least one girls is

Options

A. `1/2`
B. `1/3`
C. `2/3`
D. `4/7`

Explanation

To solve the problem, we need to find the probability that the eldest child is a girl given that there is at least one girl in the family with three children.

### Step-by-Step Solution:

1. **Define Events**:
- Let \( E \) be the event that the eldest child is a girl.
- Let \( A \) be the event that there is at least one girl in the family.

2. **Calculate \( P(A) \)**:
- The probability of having at least one girl can be calculated using the complement rule.
- The probability of having no girls (i.e., all boys) in three children is:
\[ P(\text{no girls}) = P(B) \times P(B) \times P(B) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
- Therefore, the probability of having at least one girl is:
\[ P(A) = 1 - P(\text{no girls}) = 1 - \frac{1}{8} = \frac{7}{8} \]

3. **Calculate \( P(E \cap A) \)**:
- We need to find the probability that the eldest child is a girl and there is at least one girl in the family.
- The possible combinations of children where the eldest is a girl are:
- GBB (Eldest is girl, second and third are boys)
- GBG (Eldest is girl, second is boy, third is girl)
- GGG (All three are girls)
- GGB (Eldest is girl, second is girl, third is boy)
- The total combinations satisfying \( E \cap A \) are 4:
- GBB
- GBG
- GGB
- GGG
- Each combination has a probability of \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \).
- Therefore, the probability of \( E \cap A \) is:
\[ P(E \cap A) = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

4. **Calculate \( P(E | A) \)**:
- We use the formula for conditional probability:
\[ P(E | A) = \frac{P(E \cap A)}{P(A)} \]
- Substituting the values we found:
\[ P(E | A) = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{1}{2} \times \frac{8}{7} = \frac{4}{7} \]

### Final Answer:
The probability that the eldest child is a girl given that there is at least one girl in the family is \( \frac{4}{7} \).