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The Gaming Table by Andrew Steinmetz, Volume II

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The Doctrine Of Probabilities Applied To Gambling

Among the many curious results of these inquiries according to
the doctrine of chances, is the prodigious advantage which the
repetition of odds will amount to. Thus, 'supposing I play with
an adversary who allows me the odds of 43 to 40, and agrees with
me to play till 100 stakes are won or lost on either side, on
condition that I give him an equivalent for the gain I am
entitled to by the advantage of my odds;--the question is, what I
am to give him, supposing we play at a guinea a stake? The
answer is 99 guineas and above 18 shillings,[52] which will seem
almost incredible, considering the smallness of the odds--43 to
40. Now let the odds be in any proportion, and let the number of
stakes played for be never so great, yet one general conclusion
will include all the possible cases, and the application of it to
numbers may be worked out in less than a minute's time.'[53]

[52] The guinea was worth 21s. 6d. when the work quoted was
written.

[53] De Moivre, Doctrine of Chances.

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