CHAPTER VII. 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.