The fractions which represent the probabilities of happening and failing, being added together, their sum will always be equal to unity, since the sum of their numerators will be equal to their common denominator. Now, it being a certainty that an event will either happen or fail, it follows that certainty, which may be conceived under the notion of an infinitely great degree of probability, is fitly represented by unity.
These things will be easily apprehended if it be considered that the word probability includes a double idea; first, of the number of chances whereby an event may happen; secondly, of the number of chances whereby it may either happen or fail. If I say that I have three chances to win any sum of money, it is impossible from the bare assertion to judge whether I am likely to obtain it; but if I add that the number of chances either to obtain it or miss it, is five in all, from this will ensue a comparison between the chances that are for and against me, whereby a true judgment will be formed of my probability of success; whence it necessarily follows that it is the comparative magnitude of the number of chances to happen, in respect of the whole number of chances either to happen or to fail, which is the true measure of probability.
To find the probability of throwing an ace in two throws with a single die. The probability of throwing an ace the first time is 1/6; whereof 1/ is the first part of the probability required. If the ace be missed the first time, still it may be thrown on the second; but the probability of missing it the first time is 5/6, and the probability of throwing it the second time is 1/6; therefore the probability of missing it the first time and throwing it the second, is 5/6 X 1/6 = 5/36 and this is the second part of the probability required, and therefore the probability required is in all 1/6 + 5/36 = 11/36.
To this case is analogous a question commonly proposed about throwing with two dice either six or seven in two throws, which will be easily solved, provided it be known that seven has 6 chances to come up, and six 5 chances, and that the whole number of chances in two dice is 36; for the number of chances for throwing six or seven 11, it follows that the probability of throwing either chance the first time is 11/36, but if both are missed the first time, still either may be thrown the second time; but the probability of missing both the first time is 25/36, and the probability of throwing either of them on the second is 11/36; therefore the probability of missing both of them the first time, and throwing either of them the second time, is 25/36 X 11/36 = 275/1296, and therefore the probability required is 11/36 + 275/1296 = 671/1296, and the probability of the contrary is 625/1296.
Among the many mistakes that are committed about chances, one of the most common and least suspected was that which related to lotteries. Thus, supposing a lottery wherein the proportion of the blanks to the prizes was as five to one, it was very natural to conclude that, therefore, five tickets were requisite for the chance of a prize; and yet it is demonstrable that four tickets were more than sufficient for that purpose. In like manner, supposing a lottery in which the proportion of the blanks to the prize is as thirty-nine to one (as was the lottery of 1710), it may be proved that in twenty-eight tickets a prize is as likely to be taken as not, which, though it may contradict the common notions, is nevertheless grounded upon infallible demonstrations.
When the Play of the Royal Oak was in use, some persons who lost considerably by it, had their losses chiefly occasioned by an argument of which they could not perceive the fallacy. The odds against any particular point of the ball were one and thirty to one, which entitled the adventurers, in case they were winners, to have thirty-two stakes returned, including their own; instead of which, as they had but twenty-eight, it was very plain that, on the single account of the disadvantage of the play, they lost one-eighth part of all the money played for. But the master of the ball maintained that they had no reason to complain, since he would undertake that any particular point of the ball should come up in two and twenty throws; of this he would offer to lay a wager, and actually laid it when required. The seeming contradiction between the odds of one and thirty to one, and twenty-two throws for any chance to come up, so perplexed the adventurers that they began to think the advantage was on their side, and so they went on playing and continued to lose.
The doctrine of chances tends to explode the long-standing superstition that there is in play such a thing as LUCK, good or bad. If by saying that a man has good luck, nothing more were meant than that he has been generally a gainer at play, the expression might be allowed as very proper in a short way of speaking; but if the word 'good luck' be understood to signify a certain predominant quality, so inherent in a man that he must win whenever he plays, or at least win oftener than lose, it may be denied that there is any such thing in nature. The asserters of luck maintain that sometimes they have been very lucky, and at other times they have had a prodigious run of bad luck against them, which whilst it continued obliged them to be very cautious in engaging with the fortunate. They asked how they could lose fifteen games running if bad luck had not prevailed strangely against them. But it is quite certain that although the odds against losing so many times together be very great, namely, 32,767 to 1,—yet the POSSIBILITY of it is not destroyed by the greatness of the odds, there being ONE chance in 32,768 that it may so happen; therefore it follows that the succession of lost games was still possible, without the intervention of bad luck. The accident of losing fifteen games is no more to be imputed to bad luck than the winning, with one single ticket, the highest prize in a lottery of 32,768 tickets is to be imputed to good luck, since the chances in both cases are perfectly equal. But if it be said that luck has been concerned in the latter case, the answer will be easy; for let us suppose luck not existing, or at least let us suppose its influence to be suspended,—yet the highest prize must fall into some hand or other, not as luck (for, by the hypothesis, that has been laid aside), but from the mere necessity of its falling somewhere.
Among the many curious results of these inquiries according to the doctrine of chances, is the prodigious advantage which the repetition of odds will amount to. Thus, 'supposing I play with an adversary who allows me the odds of 43 to 40, and agrees with me to play till 100 stakes are won or lost on either side, on condition that I give him an equivalent for the gain I am entitled to by the advantage of my odds;—the question is, what I am to give him, supposing we play at a guinea a stake? The answer is 99 guineas and above 18 shillings,(52) which will seem almost incredible, considering the smallness of the odds—43 to 40. Now let the odds be in any proportion, and let the number of stakes played for be never so great, yet one general conclusion will include all the possible cases, and the application of it to numbers may be worked out in less than a minute's time.'(53)
(52) The guinea was worth 21s. 6d. when the work quoted was written.
(53) De Moivre, Doctrine of Chances.