Explaining Bayes' Theorem

Date: 03/22/99 at 15:02:47
From: Kevin Battisfore
Subject: Bayes' Theorem

I need an easy to understand explanation of Bayes' Theorem to begin my 
math project for the math fair. What I want to illustrate is similar 
to the Monty Hall problem in that the subject chooses one door, is 
shown a second door, and has the option of switching from the first 
choice. I know the odds improve (from 1/3 to 2/3) if you switch, but 
I'm not sure how to explain this in terms of Bayes' Theorum.  I have 
seen the archived explanations of the Monty Hall problem, but I need 
the Bayesian background in terms my 7th and 8th grade classmates will 
understand.  

Thanks for your help.

Date: 03/22/99 at 17:02:53
From: Doctor Anthony
Subject: Re: Bayes' Theorem

A suitably simple example to make Bayes' theorem clear is the 
following:

You have 6 coins of the same denomination in a bag.  5 of them are 
standard coins, but the 6th is double-headed.

You take one coin at random from the bag and toss it 4 times. It comes 
up heads every time. What is the probability that you have the 
double-headed coin?

A diagram to illustrate the sample space will make the calculation 
clearer.

      Double-headed coin      Standard coin
           
         Prob = 1/6            Prob = 5/6
    -----------------------------------------------
          1/6 x 1             5/6 x (1/2)^4         4 heads turn up.
           = 1/6                 = 5/96

Now the sample space is fixed by the fact that we know that 4 heads 
turned up when the coin was tossed 4 times. 

The probability that we are in the first box shown above is therefore 
1/6 divided by the sum of the two boxes.

                                   1/6            1/6        16
  Prob(double-headed coin) =   -----------  =  ---------- = ----
                                1/6 + 5/96        7/32       21

So the probability that the double-headed coin was chosen =
16/21  =  0.7619 

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