Captain Walter Fried's Fish Notes Annex to #F 71, Screed on Delta D Counts and Colossus runs

Page 3

Tony Sale's
Codes and Ciphers

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3
the first to give the odds of the first theory.

Example. On the X12 run the settings 16, 09, give a score of 3.8 sigma
with a rival of 3.4 sigma. Thus the settings 16, 09 are 2 decibans up :
or p = 8/13, 10log10 p = -2.

(a) A (3+4)x/1x2x run gives X34 settings of 13, 20 with a score of
4.5 sigma and the nearest rival is 3.5 sigma : the decibanage in favour
(from the chart) is 15. Therefore 10log10(p*q/(1-q)) = 13, in fact the
whole story is 13 decibans up or 20 to 1 on.

(b) On the other hand, if the (3+4)x/1x2x run gives only a 4 sigma
reading with a 3.5 sigma rival, its decibanage is 6 up and the whole
story is then only 4 decibans up, or 5 to 2 on.

These methods enable one to estimate the odds on the combined
settings of X1234.

On the other hand, the X34 run may be a disappointment, and
if it is quite flat, this is evidence against the original X12 settings.
The factor lost by the X12 settings is a function of the probability
of the correct answer, being below the observed highest answer. This
will vary with the link and the run used : experience shows that on
a Berlin end or a Paris Jelly or Salonica Cod, a reputable score occurs
about half the time - say a score of 3.5 sigma or over. So the original
hypothesis loses a factor of 2 if the (3+4)x/1x2x run has no score above
3.5 sigma. On Rome Bream or Zagreb Gurnard on the run 4=5=/1=2,
the factor is probably in the region of 5.

Two modifying considerations.

(i) The length of the message. The chart is based on a standard length
message and on the probability of the right score being in the various
ranges. For a very short message the right score cannot be expected to
be so high and the chart is over severe; conversly for long messages it
is lenient. A standard length might be taken to be 3600.

(ii) Slides. The highest score may well be correct on one wheel and
wrong on the other. Wheels often have good slides - (i.e. two different