General Report on Tunny

23X Page 107

We now ask "What is the probability that a setting S which is s positions behind a setting T, of Ψ₁, will have coalesced with it after m BM dots ?". The question can be tied up with a problem which was stated by Lagrange. (See Uspensky "Mathematical Theory of Probability", ch 8 pp. 154, 158).

"Players A and B agreed to play not more than n games, the probabilities of winning being p and q respectively. Assuming that the fortunes of the A and B amount to a and b single stakes, find the probability for A to be ruined in the course of n games.

"The chance of A being ruined is

The first term should be replaced by if p = q = ½"

If we imagine two games played corresponding to every motor dot and equate a difference of 1 in the Ψ₁ setting to two units of the stake we can apply Lagranges result with n = 2m, p = q = ½ and a = 2s, a + b = 2 x 43 = 86. We see then that the chance that a particular Ψ₁ setting will have caught up with the setting s places ahead on the Ψ₁ wheel, after m BM dots is (if t = s/43),

If m > 500 the error involved here is very small. Thus the probability that the correct setting will have coalesced with a proportion t of the Ψ₁ settings following it, or also a proportion 1-t behind it is

so the chance of not doing this is

If m is at all large this probability is surprisingly insensitive to the size of t. Our result can be stated in the crude form:

The chance that the right setting will not have collected a high proportion of the Ψ₁ wheel, after m BM dots is roughly 1.3e^-m/750.

For a more elementary and less rigorous approach to the problem of coalescence see R4, 102. There is an interesting exposition in terms of Quantum Theory methods in R5, 71.

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