Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/38

numbers taken $s$ by $s$ times, and whose denominator is the number of combinations of $n$ numbers taken similarly $s$ by $s$ times. Thus in the lottery of France, formed as is known of 90 numbers of which five are drawn at each draw, the probability of drawing a given combination is $\scriptstyle {\frac {5}{90}}$, or $\scriptstyle {\frac {1}{18}}$; the lottery ought then for the equality of the play to give eighteen times the stake. The total number of combinations two by two of the 90 numbers is 4005, and that of the combinations two by two of 5 numbers is 10. The probability of the drawing of a given pair is then $\scriptstyle {\frac {1}{4005}}$, and the lottery ought to give four hundred and a half times the stake; it ought to give 11748 times for a given tray, 511038 times for a quaternary, and 43949268 times for a quint. The lottery is far from giving the player these advantages.

Suppose in an urn $a$ white balls, $b$ black balls, and after having drawn a ball it is put back into the urn; the probability is asked that in $n$ number of draws $m$ white balls and $n-m$ black balls will be drawn. It is clear that the number of cases that may occur at each drawing is $a+b$. Each case of the second drawing being able to combine with all the cases of the first, the number of possible cases in two drawings is the square of the binomial $a+b$. In the development of this square, the square of $a$ expresses the number of cases in which a white ball is twice drawn, the double product of$a$ by $b$ expresses the number of cases in which a white ball and a black ball are drawn. Finally, the square of $b$ expresses the number of cases in which two black balls are drawn. Continuing thus, we see generally that the $n$th power of the binomial $a+b$