Understanding the Probability of Having Two Girls in a Family with Two Children

 Assume that each child who is born is equally likely to be a boy or a girl. If a family has two children, what is the probability that both are girls given that (a) the eldest is a girl, (b) at least one is a girl?

When it comes to having children, many parents may wonder about the likelihood of having boys or girls. One common assumption is that the probability of having a boy or a girl is equal, or 50%. But what about the probability of having two girls in a family with two children? In this blog post, we will explore this question and examine the probability of having two girls given different scenarios, such as the eldest child being a girl or at least one child being a girl.

Scenario 1: The Eldest is a Girl

If the eldest child in a family with two children is a girl, what is the probability that the second child is also a girl? To answer this question, we can use conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, the event is the second child being a girl, and the given event is the eldest child being a girl.

Using the formula for conditional probability: P(B|A) = P(A and B) / P(A), where B is the event of the second child being a girl and A is the event of the eldest child being a girl.

Since the probability of having a girl is 0.5, the probability of the eldest child being a girl is also 0.5. The probability of both children being girls is 0.25 (0.5 x 0.5). So, the probability of the second child being a girl given that the eldest child is a girl is 0.25/0.5 = 0.5, or 50%.

Scenario 2: At Least One is a Girl

If a family has two children and at least one of them is a girl, what is the probability that both are girls? To answer this question, we can use the formula for conditional probability again.

The probability of the second child being a girl given that at least one child is a girl is P(B|A) = P(A and B) / P(A)

P(A) is the probability that at least one child is a girl, which is 1 - P(neither child is a girl) = 1 - (0.5 x 0.5) = 0.75

P(A and B) is the probability that both children are girls, which is 0.25.

So, P(B|A) = 0.25 / 0.75 = 0.33 or 33%.

In conclusion, the probability of having two girls in a family with two children can vary depending on the scenario. If the eldest child is a girl, the probability of the second child being a girl is 50%. However, if at least one child is a girl, the probability of both children being girls is 33%. Understanding these probabilities can help parents and families make informed decisions when it comes to having children.