**From:** Lee Corbin (*lcorbin@tsoft.com*)

**Date:** Wed Jul 09 2003 - 17:28:11 MDT

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When you are 19, your intuition is clear as a bell, and you

have utter and supreme confidence in it. You have not seen,

normally, baffling counter-examples to "obvious" theorems,

or the two-envelopes problem, Monte-Hall, Simpson's Paradox,

Unexpected Hangings, etc.

That's how it was with me in 1967 when I was 19 and first

encountered Bayes' Theorem in Gnedenko's Theory of Probability

textbook. I was appalled that something so simple as the Bayes'

Formula should have a *name*. To me, it was as if someone's

name were attached to the general solution of N equations in N

unknowns.

(This is a phenomenon among the quite young; a few years ago

a brilliant mathematician I knew---still a teenager---had run

into the Borel-Cantelli lemma in probability theory, and was

in turn appalled that something so "obvious" had a name!)

Only ten or fifteen years ago did I discover the *true*

problem that the Reverend Bayes worked on, and why indeed

his solution/theorem warrants the appellation, and why the

elementary formula also should therefore be named in his honor.

Here is the sort of problem he had obtained an answer to:

In the next room you hear shouts of "I win!" alternately

with "Damn!" from one particular player, and you realize

that he is playing some sort of game with perhaps cards

or dice.

You hear "I win!" three times, and "Damn!" twice. What

is the probability that he will win the next round?

(Or, for the non-Bayesians, what do you estimate to be

the probability or likelihood he will win the next round?)

I *believe* that the non-Bayesians will calculate a likelihood

curve, but the Bayesians will go right ahead and announce a

probability. (With a prior uniform distribution, I, and the

Reverend Bayes, and the Bayesians---I think---use a little

calculus to get an answer. I will go dig up my answer and

post it.)

Here is Bayes' original problem, in his words, in the 1750s or so:

*Given* that the number of times in which an unknown event has

happened and failed: *Required* the chance that the probability

of its happening in a single trial lies somewhere between any

two degrees of probability that can be named.

Already he wanted the probability density (distribution function).

And in his paper published later by the Royal Society, he got it.

Lee

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