Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/176

166 of $1⁄4$, of $2⁄4$, and of $3⁄4$ in the first, the second, and the third of the preceding suppositions concerning the colors of the balls in the urn. Multiplying respectively each power by the probability of the corresponding supposition, or by $3⁄6$, $2⁄6$, and $1⁄6$, the sum of the products will be the probability of extracting successively four black balls. One has thus for this probability $29⁄384$, a fraction greater than $1⁄14$. This paradox is explained by considering that the indication of the superiority of white balls over the black ones at the first drawing does not exclude at all the superiority of the black balls over the white ones, a superiority which excludes the supposition of the equality of the colors. But this superiority, though but slightly probable, ought to render the probability of drawing successively a given number of black balls greater than in this supposition if the number is considerable; and one has just seen that this commences when the given number is equal to four. Let us consider again an urn which contains several white and black balls. Let us suppose at first that there is only one white ball and one black. It is then an even bet that a white ball will be extracted in one drawing. But it seems for the equality of the bet that one who bets on extracting the white ball ought to have two drawings if the urn contains two black and one white, three drawings if it contains three black and one white, and so on; it is supposed that after each drawing the extracted ball is placed again in the urn.

We are convinced easily that this first idea is erroneous. Indeed in the case of two black and one white ball, the probability of extracting two black in two drawings is the second power of $2⁄3$ or $4⁄9$; but this