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:
P(o | h1) = 1.0
P(o | h2) = 1.0
P(o | h3) = 0.0
P(o | h4) = 1.0
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.
Hi, I am relatively new to this site, I am not sure if this is the right place to be posting.
I am sure many of you are familiar with the following probability riddle:
"Sarah is walking along the street when she encounters a man. With the man is his son. He tells Sarah that he has only one more child at home. She is asked, 'what is the probability that my child is a girl?'"
Since Sarah does not know whether the boy is the elder or younger sibling, she needs to take four possible states into account. The father either had:
1) a boy, then a girl
2) a girl, then a boy
3) two girls
4) two boys
Since 3 is impossible (Sarah knows there is at least one boy) that leaves three options. Two of those options imply a girl, the other implies a boy. Therefore, she can conclude that her probability estimate must be that it is 66.6% likely that there is a girl at home, and 33.3% likely that there is a boy.
Compare this to George's situation.
"George is walking along the street when he encounters a man. With the man is his son. He tells George that the boy with him is his oldest son, and that he has only one more child at home. He is asked, 'What is the probability that my child at home is a girl?'"
George's probability estimate is clear: either the man had a boy then a girl, or he had two boys. Therefore, it is 50% likely that the child at home is a girl.
My problem is this: I understand probability exists in the mind. The actual answer to the question is 100% one way or the other. Still, it seems like Sarah knows more about the situation, where George, by being given more information, knows less. His estimate is as good as knowing nothing other than the fact that the man has a child which could be equally likely to be a boy or a girl.
If the reply is something like "Well, Sarah actually knows less so her estimate is less likely to be right" then that is something she could have figured out on her own, and then realized that assigning probability .5 is best anyways. That seems wrong.
I know I must be making a mistake somewhere: why does it seem like George learns less by knowing more?
Thank you for your help.