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Gold/Mining/Energy : Gold and Silver Juniors, Mid-tiers and Producers
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To: emrs1 who wrote (13131)6/11/2006 2:27:47 AM
From: Gib Bogle   of 78419
 
There is an explanation along the lines of EC's post at
mathforum.org

I'd like to invite Mr Bayes into the discussion.

First, some terminology. P(A | B), where A and B are events, means: "The probability of event A given that event B occurs"
You can convince yourself that the probability of both A and B occurring, written P(AB), is P(A|B).P(B) and also that P(AB) = P(B|A).P(A).

This leads directly to the Bayes rule:
P(A | B) = P(B | A).P(A)/P(B)

Now I reckon I can use the Bayes rule in this situation.

We are interested to know, in the case that I had guessed door #1, and door #2 opens, what is the probability that the Caddie is behind door #1?

In Bayes rule, let
A = The Caddie is behind door #1
B = Door #2 opens

then we know that

P(Caddie is door #1 | Door #2 opens)
= P(Door #2 opens | Caddie is door #1)*P(Caddie is door #1)/P(Door #2 opens)

We can evaluate these 3 terms.

P(Door #2 opens | Caddie is door #1) = 1/2 (I'm assuming here that Monty Hall is equally likely to open doors 2 and 3 in this situation).

P(Caddie is door #1) = 1/3 (this is the a priori probability)

P(Door #2 opens) = ? This is a bit harder. We want the probability that door #2 opens, given only the information that I guessed #1. We have to look at the 3 possibilities for where the Caddie is.
Caddie is door #1 --> Door #2 opens with probability = 1/2
Caddie is door #2 --> Door #2 doesn't open (prob = 0)
Caddie is door #3 --> Door #2 opens with probability = 1
Combining these 3 probs, each weighted equally by 1/3, gives
(1/2 + 0 + 1)/3 = 1/2

Now we can determine P(Caddie is door #1 | door #2 opens)
= 1/2 * 1/3 / (1/2) = 1/3

In other words, the probability that the Caddie isn't door #1 (i.e. that it is door #3) = 2/3. Luckily this is the same number that EC came up with.

Was this helpful? Probably not.
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