_CONCERNING THE UNKNOWN INEQUALITIES WHICH MAY EXIST AMONG CHANCES WHICH
ARE SUPPOSED EQUAL._
Inequalities of this kind have upon the results of the calculation of
probabilities a sensible influence which deserves particular attention.
Let us take the game of heads and tails, and let us suppose that it is
equally easy to throw the one or the other side of the coin. Then the
probability of throwing heads at the first throw is ½ and that of
throwing it twice in succession is ¼. But if there exist in the coin an
inequality which causes one of the faces to appear rather than the other
without knowing which side is favored by this inequality, the
probability of throwing heads at the first throw will always ½; because
of our ignorance of which face is favored by the inequality the
probability of the simple event is increased if this inequality is
favorable to it, just so much is it diminished if the inequality is
contrary to it. But in this same ignorance the probability of throwing
heads twice in succession is increased. Indeed this probability is that
of throwing heads at the first throw multiplied by the probability that
having thrown it at the first throw it will be thrown at the second; but
its happening at the first throw is a reason for belief that the
inequality of the coin favors it; the unknown inequality increases,
then, the probability of throwing heads at the second throw; it
consequently increases the product of these two probabilities. In order
to submit this matter to calculus let us suppose that this inequality
increases by a twentieth the probability of the simple event which it
favors. If this event is heads, its probability will be ½ plus 1/20, or
11/20, and the probability of throwing it twice in succession will be
the square of 11/20, or 121/400. If the favored event is tails, the
probability of heads, will be ½ minus 1/20, or 9/20, and the probability
of throwing it twice in succession will be 81/400. Since we have at
first no reason for believing that the inequality favors one of these
events rather than the other, it is clear that in order to have the
probability of the compound event heads heads it is necessary to add the
two preceding probabilities and take the half of their sum, which gives
101/400 for this probability, which exceeds ¼ by 1/400 or by the square
of the favor 1/20 that the inequality adds to the possibilities of the
event which it favors. The probability of throwing tails tails is
similarly 101/400, but the probability of throwing heads tails or tails
heads is each 99/400; for the sum of these four probabilities ought to
equal certainty or unity. We find thus generally that the constant and
unknown causes which favor simple events which are judged equally
possible always increase the probability of the repetition of the same
simple event.
In an even number of throws heads and tails ought both to happen either
an even number of times or odd number of times. The probability of each
of these cases is ½ if the possibilities of the two faces are equal; but
if there is between them an unknown inequality, this inequality is
always favorable to the first case.
Two players whose skill is supposed to be equal play on the conditions
that at each throw that one who loses gives a counter to his adversary,
and that the game continues until one of the players has no more
counters. The calculation of the probabilities shows us that for the
equality of the play the throws of the players ought to be an inverse
ratio to their counters. But if there is between the players a small
unknown inequality, it favors that one of the players who has the
smallest number of counters. His probability of winning the game
increases if the players agree to double or triple their counters; and
it will be ½ or the same as the probability of the other player in the
case where the number of their counters should become infinite,
preserving always the same ratio.
One may correct the influence of these unknown inequalities by
submitting them themselves to the chances of hazard. Thus at the play of
heads and tails, if one has a second coin which is thrown each time with
the first and one agrees to name constantly heads the face turned up by
the second coin, the probability of throwing heads twice in succession
with the first coin will approach much nearer ¼ than in the case of a
single coin. In this last case the difference is the square of the small
increment of possibility that the unknown inequality gives to the face
of the first coin which it favors; in the other case this difference is
the quadruple product of this square by the corresponding square
relative to the second coin.
Let there be thrown into an urn a hundred numbers from 1 to 100 in the
order of numeration, and after having shaken the urn in order to mix the
numbers one is drawn; it is clear that if the mixing has been well done
the probabilities of the drawing of the numbers will be the same. But if
we fear that there is among them small differences dependent upon the
order according to which the numbers have been thrown into the urn, we
shall diminish considerably these differences by throwing into a second
urn the numbers according to the order of their drawing from the first
urn, and by shaking then this second urn in order to mix the numbers. A
third urn, a fourth urn, etc., would diminish more and more these
differences already inappreciable in the second urn.