The textbook fell into the common mistake of stating a standard probability problem with insufficient clarity. Textbooks are unfortunately especially prone to this because they crib a lot from each other, and it is very easy to miss an important condition. Unfortunately for the textook, it seems to have messed up the important condition very badly. More on that in a second.

Consider the following situation: You meet someone and ask how many kids she has. She says 2. You ask whether she has a son. She says yes. Assuming that she told the truth, what are the odds that she has a daughter? That question is stated unambiguously. And the reasoning is exactly what is in the textbook, and the answer comes out at 2/3. (Actually it is slightly below 2/3, children are slightly more likely to be male than female.)

Now consider the following alternative. You meet someone and ask how many kids she has. She says 2. You ask her for a random recent story involving one of them and she tells you something about her son Ralph. What are the odds that she has a daughter? That question is stated unambiguously as well, but the correct reasoning is very different. There is no reason to believe that she'd be more likely to come up with a story involving her daughter or her son, and now your reasoning applies and the odds come out at 50%.

Unfortunately for the textbook, the way that it stated the problem is the latter situation. In this case you are right and "the textbook answer" is wrong.

The moral is that when you state a probability problem, you need to think very hard about how you are stating it and be sure to specify not just what did happen, but what could have happened. When you ask the mother a direct question about whether she has a son and she says, "yes", you've eliminated the 1 chance in 4 that she had 2 daughters. When you ask the mother for a story, you've eliminated the 1 chance in 2 that she'll give you a story about a daughter, and that extra 1 chance in 4 that you eliminate is the possibility that she has a son and a daughter, and chose to tell you about the daughter instead of the son.

The questions were different. The possible answers were different. And therefore the appropriate probability calculations are different.

In reply to Re^5: Marilyn Vos Savant's Monty Hall problem (odd odds) by tilly
in thread Marilyn Vos Savant's Monty Hall problem by mutated

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