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