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/
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