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Simple Probability Problem

Posted By: Rick Janowski
Date: Tuesday, 25 December 2012, at 6:25 a.m.

In Response To: Simple Probability Problem (Jake Jacobs)

The probability question is very interesting and is one of the fundamental issues underpinning Elo rating systems. Most chess and backgammon Elo systems use the "logistic curve" which allows a simple equation to be formed. It assumes if A's odds again B are x to 1 and B's odds against C are y to 1 then A's odds against C are the product xy to 1. Very similar results will be obtained by using the normal distribution curve, which FIDE uses I believe. The two statistical distributions are similar, except the tails at the extremities of the normal distribution are longer. If the field is C then x and y both equal 3, so the odds appear to be 9 to 1 or 90% probability, using the logistic curve.

If the normal distribution is used in this example, I understand the approach is to calculate how many standard deviations would correspond to the probability of A beating B (= 0.75 and 0.674 standard deviations). Similarly calculate for B playing C (again 0.75 probability and 0.674 standard deviations). The probability of A beating C then can be calculated by assuming a total standard deviation equal to the sum of the two standard deviations calculated = 1.348. This corresponds to a probability of 0.911.

I suppose other statistical distributions could also be used.

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