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

### Page 11 Tony Sale's
Codes and Ciphers

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Appendix p1

Note on Odds of the result of a second run, based on a previous run.

A run has been done for X12 giving a result. "This result is correct"
we call A.

A second run, using the result of the X12 run, has been done for X34,
say (3+4)x/1x2x, giving a certain set of scores. The fact that these
scores were obtained we call z, and the statement "the highest score in
this run gives the correct setting" we call B.

P(X,Y) means "the probability of X given Y",

O(X,Y) means "the odds on X given Y",

nA means "not A"

We. make use of the theorem

(1) P(X,Y).P(Y) = P(X&Y)

We use the deciban chart after the first run to flnd P(A) = p, and after
the second run to find O(B, Z&A) = o.
(AES Note: I have used "d" in place of lower case Greek Delta, "e" in place of
epsilon and "m" in place of )

Let P(z,A) = d(1+o), P(z & nB, nA) = d' , P(z, nA) = d'(1+)

Then, by a slight extension of (1),

P(Z & B, A) = od

P(Z & nB, A) = d
and
P(Z & B, nA) = ed'

P(Z & nB, nA) = d'
Hence
P(Z) = P(Z & nA) + P(Z & A)

= P(Z, nA).P(A) + P(Z ,A)P(A)
= d(1+e)(1-p) + d(1+o)p
and
P(A & B,Z) = P(A & B & Z)/P(Z)
= P(Z & B, A).P(A) / P(Z)
= pod/P(Z)
or
O(A & B, Z) = pod/P(Z) - pod
= pod/d'(1+e)(1-p) + dp

i.e. putting d'/d = 1-m