Bayes Theorem

Date: 02/06/99 at 13:31:41
From: Brett Rogers
Subject: Bayes Theorem

This problem has caused a lot of debate at my office. Could you please 
explain the mathematical difference between the following two 
situations:

1) A woman has two children. At least one of them is a girl. What is 
   the probability that her other child is a boy?

2) A woman has two children. She is standing in front of you with her 
   10-year-old daughter. What is the probability that her other child 
   is a boy?

Thank you for your time.

Respectfully,
Brett Rogers

Date: 03/04/99 at 18:53:23
From: Doctor Stacey
Subject: Re: Bayes Theorem

A woman has two children.  Let us assume that the probability of a 
girl is one-half:  P(g) = .5

Now, what are the possibilities?

P(g,g)=.25
P(g,b)=.25
P(b,g)=.25
P(b,b)=.25

Then, in the first case, the situation becomes narrowed; we only look 
at those results where there is at least one girl. Now, given that, we 
can have (g,g), (g,b), or (b,g). The probability of a boy, given that 
there is at least one girl, is 2/3.

Now let us look at the next situation. The woman's 10-yr-old daughter 
is right there. So we know that we cannot have both (g,b) and (b,g) 
be possibilities. We do not know whether this girl is the first or 
second child, but she has to be one or the other. So, we have 
possibilities of either (g,g), (g,b), or (g,g), (b,g). In either case, 
the probability that the other child is a boy is 1/2.

I hope this helps you out. Feel free to write back if you need more 
assistance.

- Doctor Stacey, The Math Forum
  http://mathforum.org/dr.math/