CHAPTER V. CONCERNING THE ANALYTICAL METHODS OF THE CALCULUS OF PROBABILITIES.
The application of the principle which we have just expounded to the various questions of probability requires methods whose investigation has given birth to several methods of analysis and especially to the theory of combinations and to the calculus of finite differences.
If we form the product of the binomials, unity plus the first letter, unity plus the second letter, unity plus the third letter, and so on up to n letters, and subtract unity from this developed product, the result will be the sum of the combination of all these letters taken one by one, two by two, three by three, etc., each combination having unity for a coefficient. In order to have the number of combinations of these n letters taken s by s times, we shall observe that if we suppose these letters equal among themselves, the preceding product will become the nth power of the binomial one plus the first letter; thus the number of combinations of n letters taken s by s times will be the coefficient of the sth power of the first letter in the
development in this binomial; and this number is obtained by means of the known binomial formula.
Attention must be paid to the respective situations of the letters in each combination, observing that if a second letter is joined to the first it may be placed in the first or second position which gives two combinations. If we join to these combinations a third letter, we can give it in each combination the first, the second, and the third rank which forms three combinations relative to each of the two others, in all six combinations. From this it is easy to conclude that the number of arrangements of which s letters are susceptible is the product of the numbers from unity to s. In order to pay regard to the respective positions of the letters it is necessary then to multiply by this product the number of combinations of n letters s by s times, which is tantamount to taking away the denominator of the coefficient of the binomial which expresses this number.
Let us imagine a lottery composed of n numbers, of which r are drawn at each draw. The probability is demanded of the drawing of s given numbers in one draw. To arrive at this let us form a fraction whose denominator will be the number of all the cases possible or of the combinations of n letters taken r by r times, and whose numerator will be the number of all the combinations which contain the given s numbers. This last number is evidently that of the combinations of the other numbers taken n less s by n less s times. This fraction will be the required probability, and we shall easily find that it can be reduced to a fraction whose numerator is the number of combinations of r 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 5 ⁄ 90 , or 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