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then the number of figures scored out we must ,double the number of successes; to get the number added we take simply the number of failures, and the total number of sums under M is therefore the original number set under M, increased by the number of failures. He will therefore wipe out, as it were, the whole column, so soon as twice the number of successes either equals or exceeds by one the number of failures (including the first which starts the cycle). Manifestly the former sum will equal the latter, when the last win removes two numbers under M, and will exceed the latter by one when the last win removes only one number under M.
Underlying, then, the belief that this method is a certain way of increasing the gambler's store, there is the assumption that in the long run twice the number of successes will equal the number of failures, together with the number of sums originally placed under M, or with this number increased by unity. And this belief is sound; for according to the doctrine of probabilities, the number of successes--if the chances are .originally equal--will in the long run differ from the number of failures by a number which, though it may perchance be great in itself, will certainly be very small · compared with the total number of trials. So that twice the number of successes will differ very little relatively from twice the number of failures, when both numbers are large; and all that is required for our gambler's success is that twice the number of successes should equal once the number of failures, together with
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