In books on Medical Statistics
the answer to the question is stated in a mathematical formula, called
Poisson's formula, which, in a modified form, I shall give further on.
But this did not satisfy me, because I wanted to learn what a reasonably
safe _limit of error_ actually meant, and this could be best learnt by
experiment; so with the help of some friends I went in for a thorough
course of penny-tossing.
Tossing a penny twenty times, an average result would be ten heads and
ten tails. To find the deviations from this, we tossed two hundred
twenties, _i.e._, four thousand times. Of the two hundred, thirty-three
gave the exact average, viz.:--10 heads; sixty-four gave an error of
one, viz.:--9 or 11 heads; forty-nine, an error of two; twenty-six, an
error of three; twenty, an error of four; eight gave an error of five,
and this limit was not exceeded. From these we may say that six is a
reasonably safe limit of error. Ninety-seven cases, say one-half, gave
an error not exceeding one; and the mean error is 1.8.
In other words, in twenty tosses you will not get more than 16 nor less
than 4 heads; you are as likely as not to get 9, 10, or 11 heads; and
lastly, if you lost in twenty throws all heads or tails over 10 your
average loss would be 1.8 penny, or say roughly 2d. on the twenty
throws.
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