Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/175

Rh submitted to calculus, we ought to conclude that in other matters confidence should be given only after extreme circumspection. Let us prove this by example.

An urn contains four balls, black and white, but which are not all of the same color. One of these balls has been drawn whose color is white and which has been put back in the urn in order to proceed again to similar drawings. One demands the probability of extracting only black balls in the four following drawings.

If the white and black were in equal number this probability would be the fourth power of the probability $1⁄2$ of extracting a black ball at each drawing; it would be then $1⁄16$. But the extraction of a white ball at the first drawing indicates a superiority in the number of white balls in the urn; for if one supposes in the urn three white balls and one black the probability of extracting a white ball is $3⁄4$; it is $2⁄4$ if one supposes two white balls and two black; finally it is reduced to $1⁄4$ if one supposes three black balls and one white. Following the principle of the probability of causes drawn from events the probabilities of these three suppositions are among themselves as the quantities $3⁄4$, $2⁄4$, $1⁄4$; they are consequently equal to $3⁄6$, $2⁄6$, $1⁄6$. It is thus a bet of 5 against 1 that the number of black balls is inferior, or at the most equal, to that of the white. It seems then that after the extraction of a white ball at the first drawing, the probability of extracting successively four black balls ought to be less than in the case of the equality of the colors or smaller than one sixteenth. However, it is not, and it is found by a very simple calculation that this probability is greater than one fourteenth. Indeed it would be the fourth power