probability - Another look at the two children riddle

Inspired by this question about the Monty Hall problem, here's a deeper look at another well-known counter-intuitive probability problem which is often stated in a way that leaves subtle ambiguities:

Version 1

You meet a woman, who tells you: "I have exactly two children. One of them is a girl."

Version 2

You meet a woman, who tells you: "I have exactly two children. The eldest is a girl."

Version 3

You meet a woman, who tells you: "I have exactly two children. One of them is a girl." You ask- "Could you please tell me specifically a child of yours who is a girl?", and she answers "The eldest is a girl."

The question is, in all three of these cases, what is the probability that both of the woman's children are girls. Assume:

• She only tells the truth


• She always answers any question you ask to the best of her ability


• "One of them is a girl" is to be interpreted literally. It doesn't mean "exactly one of them is a girl".

As well as just giving a numerical answer for each version, also explain any apparent contradictions between the answers, and any hidden ambiguities in the question.

Answer

Too ambiguous

I think this question is ambiguous because it doesn't specify the motivation for each woman to make her particular statement. For example, why does woman number one volunteer that one of the children is a girl? If this were the situation:

Questioner: Think of one of your children and tell me the gender.
Woman: One of the children is a girl.

Then the answer would be 1/2 because it would be equivalent to the "eldest girl" response.

But if this were the situation:

Questioner: Is at least one of your children a girl?
Woman: One of the children is a girl.

Then the answer would be 1/3.

Rewording the questions

You could remove the ambiguity by changing the scenario to be the following:

You meet three women, each of which has exactly 2 children. You ask questions which each woman honestly answers. For each woman, what are the chances that both of her children are girls?

Woman 1:

You: Is at least one of your children a girl?
Woman 1: Yes, at least one of my children is a girl.

Woman 2:

You: Is your eldest child a girl?
Woman 2: Yes, my eldest child is a girl.

Woman 3:

You: Is at least one of your children a girl?
Woman 3: Yes, at least one of my children is a girl.
You: Tell me specifically one that is a girl, eldest or youngest? If you have two girls, flip a fair coin to decide which to tell me.
Woman 3: Eldest.

Now with the situation clarified, the answers should be:

1/3, 1/2, 1/3. For Woman 3, the second question doesn't actually add any information because she definitely has a daughter, and so her saying "eldest" or "youngest" doesn't change anything. It does collapse the possibilities to MF and FF, but in the MF case, she was forced to say "eldest" with 100% probability whereas for the FF case, she had a 50/50 chance to say either youngest or eldest. So even though the FM MF and FF cases were each at 1/3 before the second question, the MF case is now twice as likely as the FF case due to the 100% vs 50% response.