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### 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.

```

```
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

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

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/
```
Associated Topics:
High School Probability

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