The number of all the anagrams possible to be formed of one word, may be found in the same manner; but it must be confessed, that, if there were a great many letters in the word, the arrangements thence resulting would be so numerous, as to require a long time to find them out.

APPLICATION OF THE DOCTRINE OF COMBINATIONS TO GAMES OF CHANCE AND TO PROBABILITIES.

Though nothing, on the first view, seems more foreign to the province of mathematics, than games of chance, the powers of analysis have, as we may say, enchained this Proteus, and subjected it to calculation: it has found means to measure the different degrees of the probability of certain events, and this has given rise to a new branch of mathematics, the principles of which we shall here explain.

In a

When an event can take place in several different ways, the probability of its happening in a certain determinate manner, is greater when, in the whole of the ways in which it can happen, the greater number of them determine it to happen in that manner. lottery, for example, every one knows that the probability of obtaining a prize is greater, according as the number of the prizes is greater, and as the whole number of the tickets is less. The probability of an event, therefore, is in the compound ratio of the number of cases in which it can happen taken directly, and of the total number of those in which it can be varied, taken inversely; consequently, it may be expressed by a fraction, having for its numerator the number of the favourable chances, and for its denominator the whole of the chances.

25

Thus, in a lottery, containing 1000 tickets, 25 of which only are prizes, the probability of obtaining a prize will be represented by 35, or; if there were 50 prizes, the probability would be double; for in that case it would be equal to ; but if the number of tickets, instead of 1000, were 2000, the probability

20

would be only one half of the former, or; and if the whole number of the tickets were infinitely great, the prizes remaining the same, it would be infinitely small, or 0; while, on the other hand, it would become certainty, and be expressed by unity, if the number of the prizes were equal to that of the tickets.

Another principle of this theory, the truth of which may be readily perceived, and which it is necessary here to explain, is as follows:

A person plays an equal game when the money staked, or risked, is in the direct ratio of the probability of winning; for to play an equal game, is nothing else than to deposit a sum so proportioned to the probability of winning, that, after a great many throws, the player may find himself nearly at par; but for this purpose, the stakes must be proportioned to the probability which each of the players has in his favour. Let us suppose, for example, that A bets against B on a throw of the dice, and that there are two chances in favour of the former, and one for the latter: the game will be equal, if, after a great number of throws, they separate nearly without any loss. But, as there are two chances in favour of A, and only one for B, after 300 throws A will have won nearly 200, and B 100. A, therefore, ought to deposit 2 and B only 1; for by these means A in winning 200 throws, will get 200; and B, in winning 100, will get 200 also. In such cases, therefore, it is said, that there is two to one in favour of A.

PROBLEM r.

In tossing up, what probability is there of throwing a head several times successively, or a tail; or, in playing with several pieces, what probability is there that they will all come up heads at one throw?

As this game is well known, it is needless here to give an explanation of it; we shall therefore proceed to analyze the problem.

1st. It is evident, that as there is no reason why a head should come up rather than a tail, or a tail than a head, the probability of one of them coming up is equal to, or an equal bet may be taken on either side.

But, if any one should bet to bring heads successively in two throws, to know what in this case ought to be staked on each side, we must observe, that all the combinations possible of head and tail, which can take place, in two successive throws with the same piece, are head, head; head, tail; tail, head; tail, tail; one of which only gives head, head. Here then there is only one case in four favourable to the person who bets to throw a head twice in succession; the probability, therefore, of this event will be only ; and he who bets that it will take place, ought to deposit only a crown, while his antagonist deposits three: for the latter has three chances of winning, whereas the former has only one. To play an equal game, the money deposited by each ought to be in this proportion.

It will be found, in like manner, that he who should bet to bring a head, for example, three times successively, would have in his favour only one of the eight combinations of head and tail, which might result from three successive throws of the same piece. The probability, therefore, of this event would be, while that of his adversary would be, and consequently, to play an equal game, he ought to bet only 1 to 7.

It is needless to go over all the other cases; for it may be readily seen that the probability of throwing a head four times successively would be only, and so on. We shall say nothing farther, therefore, on the different combinations which might result from head and tail; as in all such cases the following general rule may be employed.

When the probability of two or more individual events are known, the probability of their taking place all together may be found, by multiplying together the probabilities of those events considered individually.

Thus, the probability of throwing a head, considered individually, being expressed at each throw by, that of throwing it twice successively, will be x =

that of throwing it three times successively, will be , and so on.

2nd. The probability of throwing all heads, or all tails, with two, three, or four pieces, may be determined in the same manner. When two pieces are employed, there are four combinations of head and tail, only one of which is both heads; when three pieces are tossed up, at the same time, there are 8, one of which only is all heads; and so on. The probability, therefore, in each of these cases, is similar to those already examined.

It may be seen, indeed, without the help of analysis, that these two questions are absolutely the same, as may be proved in the following manner:-To toss up the two pieces, A and B, at the same time, or to throw up the one after the other, when A the first has had time to settle, is certainly the same thing. Let us suppose, then, that when A the first has settled, instead of tossing up B, the second, A, is taken from the ground, in order to be tossed up a second time: this will certainly be the same thing as if the piece B had been employed; for, by the supposition they are both equal and similar, at least in regard to the chance of a head or a tail coming uppermost; consequently, to toss up the two pieces A, B, at once, or to toss up twice successively the piece A, is the same thing.

3rd. If it were asked, how much a person might bet to bring a head at least once in two throws, it may be found by the above method that the chance is 3 to 1. In two throws there are four combinations, three of which give at least one head, while there is only one which gives two tails; and hence it follows, that there are three combinations in favour of the person who bets to bring a head once in two throws, and only one against him.

PROBLEM II.

Any number of dice being given, to determine the probability of throwing with them an assigned number of points.

We here suppose that the dice are of the usual kind, that is to say, having six faces marked with the num

bers 1, 2, 3, 4, 5, 6. This being premised, we shall analyse some of the first cases of the problem, that we may proceed gradually to those which are more complex. 1st. It is proposed to throw a determinate point, for example, 6, with one die.

As the die has six faces, one of which only is marked six, and as any one of these may come up as readily as another, it is evident that there are 5 chances against the person who undertakes to throw 6 at one throw, and only 1 in his favour. To play an equal game he ought, therefore, to bet no more than 1 to 5.

2nd. Let it be proposed to throw the same point 6, with two dice.

To analyze this case, it must first be observed that two dice give 36 different combinations; for each of the faces of the die A, for example, may combine with each of those of B, which will produce 36 combinations. We must next examine in how many ways the point 6 can be thrown with two dice. 1st. It will be found that it can be thrown by 3 and 3; secondly, by throwing 2 with the die A, and 4 with the die B, or 4 with A and 2 with B, which, as may be readily seen, forms two distinct cases; thirdly, by throwing 1 with A and 5 with B, or 1 with B and 5 with A, which likewise forms two cases: these are evidently all the ways that can be found. Consequently, there are 5 cases favourable in the 36; and, therefore, the probability of throwing 6 with two dice, is, and that of not throwing it; hence it appears that the money staked by the players ought to be in the ratio of these two fractions.

By analysing the other cases it will be found, that of throwing 2 with two dice, there is 1 chance in 36; of throwing 3, there are 2; of throwing 4, there are 3; of throwing 5, there are 4; of throwing 6, there are 5; of throwing 7, there are 6; of throwing 8, there are 5; of throwing 9, there are 4; of throwing 10, there are 3; of throwing 11, there are 2; and of throwing 12, or sixes, there is 1.

If three dice were proposed, with which the least point that could be thrown is evidently 3, and the