Question

There are three children in a family. If it is known that at least one child is a girl among them, find the probability that all three children are girls.

Hint:

When calculating conditional probabilities, identify the restricted set of outcomes based on the given condition and then calculate the probability based on favorable outcomes.

Answer 1

We are given that there are three children, and at least one child is a girl. We are asked to find the probability that all three children are girls, given this condition.

Step 1: Total possible outcomes.
Each child can either be a boy or a girl. Hence, for three children, the total number of possible outcomes is: \[ 2 \times 2 \times 2 = 8. \] The possible combinations of boys (B) and girls (G) are: \[ GGG, GGB, GBG, BGG, BBB, BBG, BGB, GBB. \]

Step 2: Restricting to the condition.
We are told that at least one child is a girl. Therefore, we eliminate the outcome \( BBB \) where there are no girls. This leaves us with the following 7 possible outcomes: \[ GGG, GGB, GBG, BGG, BBG, BGB, GBB. \]

Step 3: Favorable outcomes.
We want the probability that all three children are girls, which corresponds to the outcome \( GGG \).

Step 4: Calculate the probability.
The number of favorable outcomes is 1 (i.e., \( GGG \)), and the total number of possible outcomes, given that at least one child is a girl, is 7. Hence, the probability is: \[ P(\text{All girls} | \text{At least one girl}) = \frac{1}{7}. \] Conclusion: The probability that all three children are girls, given that at least one child is a girl, is: \[ \boxed{\frac{1}{7}}. \]