Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/67

Rh 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 $1⁄2$ 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 $1⁄2$ 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 by $1⁄4$ 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