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Probability that a 6.0 player can join Mochy's 4.0 club
Posted By: Timothy Chow
Date: Tuesday, 8 April 2014, at 6:58 p.m.
In December, Neil Robins asked if any quantitative justification could be provided for Mochy's intuition that a 6.0 player who tries to join Mochy's 4.0 club has less than a 1% chance of success. While this sort of thing can never be proven mathematically, I've come up with an argument that some might find interesting.
The idea is to model the distribution of a player's PR as a gamma distribution. Gamma distributions are governed by two parameters, k and θ, which can be thought of as controlling the mean and the variance of the distribution. Specifically, the product kθ of the two parameters is the mean, and for a fixed mean, increasing θ increases the variance. Other than the fact that the pictures of the gamma distribution look sort of like how PR's tend to be distributed, there is a handwaving justification: when k is an integer, the gamma distribution gives the distribution of k independent draws from an exponential distribution. So if you think that a match gives you a certain number of opportunities to err, and each error is exponentially distributed, then you expect something like a gamma distribution to emerge.
Anyway, I experimented with a few values of k and θ and found that k = 4 and θ = 3/2 seemed to be plausible parameters for a "typical" 6.0 player. With these parameters, the probability (in any given single match) of playing > 10.0 is about 10%, the probability of playing below 4.0 is about 28%, and the probability of playing below 1.0 is about 0.5%.
The bottom line is that if we now ask for the probability that ten independent random draws from the above distribution will average to less than 4.0, then the answer comes out to be just under 1%. (Of course the overall PR of ten matches is not the simple average of the ten individual PR's, but given all the other tall tales I'm telling, this one seems relatively forgivable.)

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