Lost Town and Finder's Town: Bayesian Probability

Date: 04/27/99 at 16:08:18
From: Wally Ming
Subject: Probabilities

A man is lost in the mountains. There is a town located near him 
called Lost Town that's populated with 10 males and 10 females. 
Another town that's also close is Finder's Town with a population of 
4 females and 16 males. He has no idea what town he's in, but he 
spots a male. What is the probability that he wandered into Finder's 
town? What is the probability he wandered into Lost Town?

They way I tried to solve it:

There's a 50% (10/20) chance that he meets up with a male if he 
wanders into Lost Town. There's a 80% (16/20) chance he meets a male 
if he wanders into Finder's town. 

.50(10) = 5% chance the guy is in Lost Town
.80(16) = 12.8% chance the guy is in Finder's Town

Is this right?

Date: 04/27/99 at 18:51:13
From: Doctor Anthony
Subject: Re: Probabilities

This is an example of Bayesian probability.  You are given extra 
information that changes the probabilities from those you expect 
before the experiment begins. In this case the extra information is 
that he spots a male

        Lost Town       Finder's Town

        Prob = 1/2      Prob = 1/2
    -------------------------------------
        (1/2)(1/2)      (1/2)(16/20)       He spots a male
             = 1/4           = 2/5

The sample space is confined to the sum of these two probabilities 
since he spots a male.

                                        1/4          1/4
The probability he is in Lost Town = ---------  =  -------
                                     1/4 + 2/5      13/20

                                   =  5/13

and the probability he is in Finder's Town is then 8/13.

Note that the two probabilities MUST sum to 1 since he is certainly 
in one or the other of the towns.

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