It was like a case of nine trumps, which though in a sense
possible, is very unlikely to happen in any one's experience.
But even now we are not quite in a position to answer the question with
which we started. If you refer to it you will see that we are face to
face with this problem: the limit of variation on the 1000 who died
would be say 70,[127] ignoring decimals. But if we calculate on the
number who did not die, viz.--699,000,[128] we shall get a variation 26
times as great as this. But it is evident the variation must be the same
in each case. I submitted this kind of problem also to the test of
experiment, the results of which gave me great faith in Poisson's
formula.
Imagine two hundred pennies in a bag all heads up. Any shaking will
spoil this arrangement and give a certain proportion of tails. And,
further, the probable effect of shaking and turning will be to reduce
the preponderance of heads or tails whichever may be in excess. This of
course is the reason why we are so unlikely to get more than 120 of them
in either position.
But if the two hundred pennies are increased to 20,000 by adding pennies
which have tails on both sides, then the shaking or mixing would be less
effective. We should still expect as an average result to get the 100
heads but in 20,000 instead of 200. The variation will be 28 or 29 on
the 100 instead of 20.
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