A family with two children is randomly selected. Given that the event of having a boy or a girl are equally likely. Equally likely events are events that have the same probability of occurring. Let B denote the event that “the child is a boy” and let G denote the event that “the child is a girl”.

In a two-child family, there are only four possible combinations of the children. The sample space is:

 BB, GB, BG, GG

           

It is given that the oldest child is a boy. The sample space for this is reduced to:

BB, BG

Because the two other possibilities (GB, GG) shows girl child is the oldest child and thus it is not possible.

 Since only one of the possibility in the new sample space, (BB) includes both boys; the probability that the family has two boys given that the oldest child is a boy is ½.

 The probability that the family has two boys given that the oldest child is a boy is 1/2.

similarly for other: 1/3