There are two children in a family. If it is given that at least one of them is a boy, what is the probability that both the children are boys?

Correct option is C

Step 1: Total possible outcomes

For two children, each child can either be a boy (B) or a girl (G). The total outcomes are:
$$\{BB, BG, GB, GG\}$$​
There are 4 total outcomes.

Step 2: Favorable outcomes (at least one boy)
If at least one of the children is a boy, we exclude the outcome GG . The remaining outcomes are:
BB, BG, GB
There are 3 favorable outcomes.

Step 3: Probability formula
The probability that both children are boys, given that at least one of them is a boy, is:
$$P(\text{both boys | at least one boy}) = \frac{\text{Favorable outcomes for both boys}}{\text{Total favorable outcomes}}$$​

Step 4: Calculate the probability
The only outcome where both children are boys is BB . Thus, the number of favorable outcomes is 1 :
$$P(\text{both boys | at least one boy}) = \frac{1}{3}$$​

Final Answer:

The probability that both children are boys, given that at least one of them is a boy, is:
$$\frac{1}{3}$$​