Chance and Luck

It is well to have some convenient standard of reference, not only as respects the fairness or unfairness of betting transactions, but as to the true nature of the chances involved or supposed to be involved. Many men bet on horse races without any clear idea of the chances they are really running. To see that this is so, it is only necessary to notice the preposterous way in which many bettors combine their bets. I do not say that many, even among the idiots who wager on horses they know nothing about, would lay heavier odds against the winning of a race by one of two horses than he would lay against the chance of either horse separately; but it is quite certain that not one bettor in a hundred knows either how to combine the odds against two, three, or more horses, so as to get the odds about the lot, or how to calculate the chance of double, triple, or multiple events. Yet these are the very first principles of betting; and a man who bets without knowing anything about such matters runs as good a chance of ultimate success as a man who, without knowing the country, should take a straight line in the hunting-field.

Now, apart from what may be called roguery in horse-racing, every bet in a race may be brought into direct comparison with the simple and easily understood chance of success in a lottery where there is a single prize, and therefore only one prize ticket: and the chance of the winner of a race, where several horses run, being one particular horse, or one of any two, three, or more horses, can always be compared with the easily understood chance of drawing a ball of one color out of a vase containing so many balls of that color and so many of another. So also can the chance of a double or triple event be compared with a chance of the second kind.

Let us first, then, take the case of a simple lottery, and distinguish between a fair lottery and an unfair one. Every actual lottery, I remark in passing, is an unfair one; at least. I have never yet heard of a fair one, and I can imagine no possible case in which it would be worth anyone's while to start a fair lottery.

Suppose ten persons each contribute a sovereign to form a prize of 10£; and that each of the ten is allowed to draw one ticket from among ten, one marked ticket giving the drawer the prize. That is a fair lottery; each person has paid the right price for his chance. The proof is, that if anyone buys up all the chances at the price, thus securing the certainty of drawing the marked ticket, he obtains as a prize precisely the sum he has expended.

This, I may remark, is the essential condition for a fair lottery, whatever the number of prizes; though we have no occasion to consider here any case except the very simple case of a one-prize lottery. Where there are several prizes, whether equal or unequal in value, we have only to add their value together: the price for all the tickets together must equal the sum we thus obtain. For instance, if the ten persons in our illustrative case, instead of marking one ticket were to mark three, for prizes worth 5£, 3£, and 2£, the lottery would be equally fair. Anyone, by buying up all the ten tickets, would be sure of all three prizes, that is, he would pay ten pounds and get ten pounds -- a fair bargain.