General Report on Tunny
24X Page 144
| IV. | Consider first the two theories |
| (i) Two definite wheel patterns and a definite value of δ | |
| (ii) δ = 0 (i.e. rectangle is random). |
The probability of an excess Θ of dots over crosses in a cell of the rectangle containing k entries (where Θ and k have the same parity) is
if ΔΧ12 is assumed to be a dot in the cell. Therefore the factor for theory (i) rather than (ii) is
Therefore, using all the cells of the rectangle, the total factor in favour of theory (i) rather than (ii) is
where x is the double bulge of /1+2 using the wheel patterns of theory (i). Denote by φ(δ) the prior probability distribution of δ. Then the factor in favour of the particular wheel patterns, not allowing for competition is a number f where
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where  |
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Therefore f =  |
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If+ x2 < 120N, the term x4/(12N3) is less than 10(x/N)2 (natural bans) which is nearly always negligible. If we assume δ has a uniform distribution in an interval* of length .1, and has no chance of
+ See below
* This estimate was originally a guess, but it was borne out quite well by statistics of set messages. In any case the result is not sensitive to variations in the assumption of the precise distribution of δ.
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