Which of these two women is more likely to be the parent of two girls? Explain.

Question

Professor Robertson and Professor Simpson are both mathematicians. Professor Robertson tells you that she has exactly two children, at least one of whom is a girl. Then Professor Simpson tells you she has exactly two children also, and that the older one is a girl, but says nothing about the gender of they younger child. Which of these women is more likely to be the parent of two girls? Explain.

For Professor Robertson you need to make sense of what they mean by "at least" one is a girl. where for professor Simpson, you would need to find the probability of one child is a girl and the 2nd child being a boy or a girl. then compare the two probabilities and you should be able to answer the question based on that.

Answer 1

Professor Robertson has at least one girl, so that means there are only 3 possible situations for Professor Robertson, Child A is a girl and Child B is a boy, Child A is a boy and Child B is a girl, or both children are girls. There is only one way that Professor Robertson can have two girls, so the probability that she has two girls is 1/3.

We already know that Professor Simpson has 1 girl, so then there are only two possible situations for Professor Simpson, the older child is a girl and the younger child is a boy, or both children are girls. There is only one way Professor Simpson can have two girls, so the probability that she has two girls is 1/2.

Answer 2

A different perspective:

In both cases, we are told that there is one girl.

We are not told anything about the second child.

The ages are not relevant.

Whether you say "I have one girl" or "I have at least one girl", you have not said anything about the gender of the 2nd child. To me that says the odds of the second child being a girl are 50% in both cases.