The problem with the Unfinished Game is that it does NOT distinguish properly between what Jeff Atwood intended it to be (hence ending with a 2/3 probability) and the “nosy neighbor” scenario.
It comes to the same downfall as Monty Hall/Fall: intent is KEY because it shifts the probabilities.
In Monty Fall, the probabilities shift to 50/50 because Monty is not adding information (more information is coming in, but that information is as likely to drop us out of the scenario a (he accidentally opens the door we chose, or he accidentally opens the door with the car behind it) as it is to guide us. Thus, the problem domain has contracted but no new information has been imparted on the existing choices.
In the Unfinished Game scenario, we are supposed to realize that the information offered is NOT random, but as a result of a pointed question (“Is one of your children a girl”) but not TOO pointed question (“Is your oldest child a girl”). If it were random, you end up with the “nosy neighbor” scenario, wherein the extra information contracts the problem set (you are as likely to have seen a boy walking across as a girl) instead of ellucidating the remaining domain.
Which brings me back to the first paragraph of Atwood’s piece, where he scolds us for telling him his formulation of the Unfinished Game problem was incredibly poorly stated. Sorry, but it was. He does not say if this person pointedly told us he has a girl at random or with some surrounding information, and supposes that we must infer that there was some surrounding context to that information.
Personally, as a parent, I answered “~1”. If you are a parent and tell someone that “one of my children is a girl” you are saying that the other one is not. Simple human nature. If I have two girls, I’m going to say I have two girls, not one. The only possibilities then are:
- The other child is a boy
- The other child is a hermaphrodite of some sort
- The person is being deliberately coy about their children’s genders
Since 2 and 3 are generally very small probabilities, that leaves just about a 100% chance that the other child is a boy.
Of course, Atwood’s scenario posits that (3) is the dominant probability, which then overwhelms the odds of the situation, and further that there is not a REASON that they are being deliberately coy (for instance, some odd societal stature loss from having two girls).