Question

Given a family of two children (assume boys and girls equally likely, that is, probability for each), what is the probability 1/2 that both are boys? That at least one is a girl? Given that at least one is a girl, what is the probability that both are girls? Given that the first two are girls, what is the probability that an expected third child will be a boy?

Short answer

The probability that both are boys is 1/4.

The probability that at least one is a girl is 3/4.

When at least one is a girl, the probability that both are girls is 1/3.

When the first two are girls, the probability that an expected third child will be a boy is 1/2.

Step-by-step solution

Expression for the Probability

The probability for an event E is expressed as follows,

P = number of outcomes favorable to E/total number of outcomes ...(i)

Determination of the probability that both are boys

A child being a girl or a boy does not depend on whether the previous child was a girl or a boy.

Let G represents girl child and B represents Boy child, thus the possible outcomes for two children are 4 namely (BB, BG, GB, GG).

In possible outcome, only one case namely (BB) is the favorable outcome, this implies that the probability that both are boys is 1/4.

Determination of the probability that at least one is a girl 

In possible outcome, only three cases namely (GG, GB, BG) are the favorable outcome, this implies that the probability that at least one is a girl is 3/4.

Determination of the probability that both are girls when at least one is a girl

In possible outcomes, only three cases namely (GG, GB, BG) has the at least one girl, out of which only one case namely (GG) is the favorable outcome, this implies that the probability that both are girls when at least one is a girl is 1/3.

Determination of the probability that an expected third child will be a boy given that the first two children are girls 

The probability of the third child being a boy is 1/2 as it does not depend whether the previous children were boy or girl.

Thus, the probability that both are boys is 1/4, the probability that at least one is a girl is 3/4, the probability that both are girls when at least one is a girl is 1/3, and the probability of the third child being a boy is 1/2.