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### 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/
```
Associated Topics:
High School Probability

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