Code and ciphers: Julius Caesar, the Enigma and the internet

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appendix
and, since all random keys of 15 letters are equally likely, either of these
could be correct and indeed there are more than 10 21 other possible
decrypts, most of which are, however, nonsense.
Chapter 8
M8 Frequency of occurrence in a page of random
numbers
In a page of 100 two-digit random numbers any number in the range 00
to 99 can be expected to occur once. The probability that a particular
number will not occur in any particular position is 0.99 and, since the
numbers are random the probability that any particular number will not
occur at all in a page of 100 is
(0.99) 100 or (1�1/100) 100
the value of which is, effectively, e �1 , where e�2.718 28... is the base of
natural logarithms, as was remarked before, in M1. Since e �1 �0.37 to
2 d.p. it follows that in a typical page of 100 random two-digit numbers
there will be about 37 that do not occur. On the other hand there should
be about
100e �1
which occur three times, i.e. about 6, and we might expect one number to
occur four times since the expected number in that case is
100e �1
the value of which lies between 1 and 2.
(What we are doing, in effect, is claiming that the probability of a specified
two-digit number occurring exactly n times on a page of 100 such
random numbers is approximately
e �1
n!
3!
4!
which is a particular case of what is known as the Poisson distribution in
probability theory. For a full mathematical treatment and justification,
consult books on probability theory, such as [8.1].)