Boy or girl paradox: Difference between revisions

The Boy or Girl problem is a well known example in probability theory.

Essentially it encompasses two questions

• In a two-child family, one child is a boy. What is the probability that the other child is a girl?
• What if the older child is a boy? Does this information change the probability that the second child is a girl?

This problem can be solved using combinations in the sample space.

In a two-child family, there are four and only four possible combinations of children. We will label boys B and girls G; in each case the first letter represents the oldest child:

{BB, BG, GB, GG}

When we know that one child is a boy, there cannot be two girls, so the sample space shrinks to:

{BB, BG, GB}

Two of the possibilities in this new sample space include girls:

{BG, GB}

and since there are two combinations out of three that include girls, the probability that the other child is a girl is 2/3.

The answer just given, which is the standard answer, overlooks one important thing. We are not told whether the boy is the eldest or the youngest. (Unlike in the second question.) Therefore, even once we shrink the sample space to

{BB, BG, GB}

we still don't know which of the 4 boys have been identified. The bold B shows which boy was identified:

BB, BG, GB = boy sibling
BB, BG, GB = boy sibling
BB, BG, GB = girl sibling
GG, BG, GB = girl sibling

two boys, two girls = 50%, not the 2/3 answer given above.

When the oldest child is a boy, the original sample space:

{BB, BG, GB, GG}

shrinks to:

{BB, BG}

because the two other possibilities, {GB, GG}, show girls as the oldest child and thus are no longer possible.

Since only one of the possibilities in the new sample space, {BG}, includes a girl, the probability that the second child in the family is a girl is 1/2.

• Three cards problem