In article <firstname.lastname@example.org>, quasi <email@example.com> wrote:
> On Tue, 11 Mar 2008 21:56:27 EDT, Steven <firstname.lastname@example.org> > wrote: > > >Suppose you meet me on a street corner and I introduce you to my son who is > >with me. I inform you that I have another child at home. What is the > >probability that my other child is a girl. > > The problem is not adequately specified. > > It depends on how the child accompanying the father is selected. > > If the child that accompanies the father is selected at random by a > flip of a fair coin, then the probability that the other child is a > girl is 1/3.
The sample space for the children is (b b), (b g), (g b), (g g) where the first slot is the youngest child, the second slot is the oldest. These oredered pairs all have probability 1/4. Now we select a child at random for a walk. We get a new sample space: (b b b), (b g g), (b g b), (g b g), (g b b), (g g g), with the probabilities being 1/4 for the first and last triples, and 1/8 for the others. The probability the other child is a girl given the randomly selected child out with daddy is a boy is thus
p((b g b) (g b b))/p((b b b) (b g b) (g b b))
= (1/8 + 1/8)/(1/4 + 1/8 + 1/8) = 1/2.
> If the father always chooses the younger child to go with him, then > there is no information, so the probability that the other child is a > girl is 1/2. > > Similarly, if the father always chooses the older child to go with > him, then again there is no information, so the probability that the > other child is a girl is 1/2. > > If a boy has _priority_ over a girl to go with the father, then the > probability that the other child is a girl is 2/3. > > If a girl has _priority_ over a boy to go with the father, then the > probability that the other child is a girl is 0. > > Thus, the probability critically depends on the selection mechanism. > > In the absence of any information, the problem is not well posed. > > Of course, in a given situation. you can always make a subjective > judgement as to the type of selection, in which case, you can then > derive an answer based on that extra assumption. > > For example, thinking about it culturally, if I _had_ to guess, I > would guess that the father would choose to take the _younger_ child, > in which case, the probability is 1/2. > > quasi