The standard formulation of the problem is such you are the one making the bizarre contortions of conditional probabilities by asking a question. The standard setup has no children with the person you meet, he tells you only that he has two children, and you ask him a question rather than them revealing information. When you ask “Is at least one a boy?”, you set up the situation such that the conditional probabilities of various responses are very different.
In this new experimental setup (which is in very real fact a different problem from either of the ones you posed), we end up with the following situation:
h1 = "Boy then Girl"
h2 = "Girl then Boy"
h3 = "Girl then Girl"
h4 = "Boy then Boy"
o = "The man says yes to your question"
With a different set of conditional probabilities:
And it’s relatively clear just from the conditional probabilities why we should expect to get an answer of 1⁄3 in this case now (because there are three hypotheses consistent with the observation and they all predict it to be equally likely).
The standard formulation of the problem is such you are the one making the bizarre contortions of conditional probabilities by asking a question. The standard setup has no children with the person you meet, he tells you only that he has two children, and you ask him a question rather than them revealing information. When you ask “Is at least one a boy?”, you set up the situation such that the conditional probabilities of various responses are very different.
In this new experimental setup (which is in very real fact a different problem from either of the ones you posed), we end up with the following situation:
With a different set of conditional probabilities:
And it’s relatively clear just from the conditional probabilities why we should expect to get an answer of 1⁄3 in this case now (because there are three hypotheses consistent with the observation and they all predict it to be equally likely).
That makes a lot of sense, thank you.