Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that i. the youngest is a girl, ii. at least one is a girl?

Explanation

To solve the problem, we will break it down into two parts as specified in the question. We will first determine the sample space and then calculate the conditional probabilities for each scenario.

### Step 1: Define the Sample Space
When a family has two children, each child can either be a boy (B) or a girl (G). Therefore, the sample space (S) for the genders of the two children can be represented as follows:
- BB (both are boys)
- BG (first is a boy, second is a girl)
- GB (first is a girl, second is a boy)
- GG (both are girls)
...