_GAMES OF CHANCE._
The combinations which games present were the object of the first
investigations of probabilities. In an infinite variety of these
combinations many of them lend themselves readily to calculus; others
require more difficult calculi; and the difficulties increasing in the
measure that the combinations become more complicated, the desire to
surmount them and curiosity have excited geometricians to perfect more
and more this kind of analysis. It has been seen already that the
benefits of a lottery are easily determined by the theory of
combinations. But it is more difficult to know in how many draws one can
bet one against one, for example that all the numbers will be drawn, _n_
being the number of numbers, _r_ that of the numbers drawn at each draw,
and _i_ the unknown number of draws. The expression of the probability
of drawing all the numbers depends upon the _n_th finite difference of
the _i_ power of a product of _r_ consecutive numbers. When the number
_n_ is considerable the search for the value of _i_ which renders this
probability equal to ½ becomes impossible at least unless this
difference is converted into a very converging series. This is easily
done by the method here below indicated by the approximations of
functions of very large numbers. It is found thus since the lottery is
composed of ten thousand numbers, one of which is drawn at each draw,
that there is a disadvantage in betting one against one that all the
numbers will be drawn in 95767 draws and an advantage in making the same
bet for 95768 draws. In the lottery of France this bet is
disadvantageous for 85 draws and advantageous for 86 draws.
Let us consider again two players, A and B, playing together at heads
and tails in such a manner that at each throw if heads turns up A gives
one counter to B, who gives him one if tails turns up; the number of
counters of B is limited, while that of A is unlimited, and the game is
to end only when B shall have no more counters. We ask in how many
throws one should bet one to one that the game will end. The expression
of the probability that the game will end in an _i_ number of throws is
given by a series which comprises a great number of terms and factors if
the number of counters of B is considerable; the search for the value of
the unknown _i_ which renders this series ½ would then be impossible if
we did not reduce the same to a very convergent series. In applying to
it the method of which we have just spoken, we find a very simple
expression for the unknown from which it results that if, for example, B
has a hundred counters, it is a bet of a little less than one against
one that the game will end in 23780 throws, and a bet of a little more
than one against one that it will end in 23781 throws.
These two examples added to those we have already given are sufficient
to shows how the problems of games have contributed to the perfection of
analysis.