In the spot-the-girl-on-the-lawn scenario, or the girl-comes-to-door scenario, there is an extra random event (beyond the original births). One child of the family is selected at random to come to the door, or to go out on the lawn. That changes the odds.

Anyhow, it’s a new week; no more on this .

Then someone mentions that at least one of the children is a girl. So, you say to yourself “Too bad, but hey, that still leaves a 66% chance the other child is a boy.”

The neighbors move in and you spot one of their children on the lawn. It is a girl. You tell yourself “That’s okay, I know one of them is a girl, so the probability of a boy is still 66%: she could have an older brother, a younger brother, or a sister.” But oops, that list is incomplete, because she could in fact have an older brother, a younger brother, an older sister, or a younger sister. There are four possibilities. Or, if you remove the now unnecessary question of younger versus older, there are two possibilities: she has a sister or she has a brother. Thus, you now calculate the probability that her sibling is a boy at 50%.

But you already knew one of the children is a girl. So, how does spotting a girl on the neighbor’s lawn reduce the chances of your son getting a boy to play with?

The pat answer would be you are considering two different scenarios, asking two different questions. In the first scenario, you are considering the gender distribution within a randomly selected two-child family (looking at the possible permutations of two variables: gender of child #1 and gender of child #2). In the second scenario, when you spot the girl on the lawn, you are asking “what is the probability her sibling is a boy?” There is only one variable in that question (the gender of the other child). I don’t see a bridge between the two scenarios, and I think that is part of what leads to the enduring controversy from this form of question. And underlines the need for carefully wording the question.

Sorry. Talking about a specific child at the door raises a bunch of different considerations (and I typed girl instead of boy). The question should have been “Your neighbor has two children, one of which is a girl. What are the odds the other child is a boy?”

However, by randomly choosing one of the children and naming that child as a girl, the answer becomes 50%. You are effectively asking the following question: “The girl at the door has one sibling. What is the probability the girl’s sibling is a boy?” Questions of relative birth rates for boys and girls aside, I assure you the answer is 50%. It doesn’t matter if the girl is older or younger sibling. She has one sibling. There is a 50% chance the sibling is a boy and a 50% chance the sibling is a girl. If the phrasing of the question reduces things to the consideration of one child, the answer is 50%. If the phrasing is such that you need to consider both children, the answer is 66%.

Based on the phrasing of the question posed here, Dave is in fact correct.

“The second question is often posed in a way that leave multiple interpretations open. In response to reader criticism of the question posed in 1959, Gardner agreed that a precise formulation of the question is critical to getting different answers for question 1 and 2[1]. Specifically, Gardner argued that a “failure to specify the randomizing procedure” could lead readers to interpret the question in two distinct ways[1]:

* From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.

* From all families with two children, one child is selected at random, and the gender of that child is specified. This would yield an answer of 1/2, and many experts agree.[3][4]

The ‘comes to the door’ case corresponds to the second case.

“However, if it is assumed that the information was obtained by considering only one child, then the problem is an isomorphism of question one, and the answer is 1/2.”

This is the case when a child comes to the door – you’re just picking one of the children and considering them (a random act).

As long as you don’t know wcich child you’re looking at, the odds are 2/3

http://en.m.wikipedia.org/wiki/Boy_or_Girl_paradox