In order to illustrate the solution, we choose an urn model. All families are considered, but we are interested in only those with two children. The urn model is limited to only those families with two children. These children could have been born in the following orders:
- Girl, Girl (GG)
- Girl, Boy (GB)
- Boy, Boy (BB)
- Boy, Girl (BG)
As the probability of giving birth to a girl or a boy is equal, the same number of each of these four family varieties is represented in the urn. Supposing
- 10,000 families with two daughters
- 10,000 families getting first a daughter, and then a son
- 10,000 families with two sons
- 10,000 families getting first a son, and then a daughter
Since we already know that Mrs Schmidt has a daughter, we take all families with two boys out of the urn. Now, it only contains families of the varieties GG, BG and GB – each variety is represented with the same number. In our example this means 10,000 families with two girls and a total number of 20,000 families having a son and a daughter. Therefore, the probability that Mrs Müller has two daughters is one-third.
Conversely, with a two-thirds probability, Mrs Müller has a son and a daughter.