In the first place, by simply doubling up you are giving the bank the best of it, because you are not getting the proper odds. If you double up five times you are betting 16 to 1; but the odds against five successive events are 31 to 1, as we have already seen, and the bank should pay you 31 instead of 16. You should not only double, but add the original amount of the stake each time, betting 1, 3, 7, 15, 31, 63, and so on. If you do this, you will win the amount of your original stake for every bet you make, instead of only for every time you win. This looks well, but as a matter of fact doubling up is only another way of borrowing small sums which will have to be paid back in one large sum when you can probably least afford it.

Suppose the game is Faro, the chips five dollars a stack, and the limit on cases twenty-five dollars. The limit on cases will then be 400 chips. If eight successive events go against your “system,” which they will do about once in 255 times, your next bet will be beyond the limit, and the banker will not accept it. At Monte Carlo the smallest bet is a dollar, and the limit is $2,400. They roll about 4,000 coups a week, and if you were to bet on every one of them, doubling up, you would win about $1,865, one dollar at a time, and would lose $4,092 simply through being unable to follow your system beyond the limit of the game during the two or three occasions, in the 4,000 coups, that your system would go against you for eleven or more coups in succession. It is useless to say it would not go against you so often, for probabilities teach us that it would be more wonderful if it did not than if it did.

It must never be forgotten that the most wonderful things that happen are not more wonderful than those that don’t happen. If you tossed a coin a thousand times, and did not once toss heads eight times in succession, it would be four times more surprising than if you tossed heads ten times in succession.

Bets
Won.Lost.
10-
9-
8-
-7
-8
9-
-8
-9
10-
-9
4641

Progression. This is a favourite martingale with those who have not the courage or the money to double up. It consists in starting with a certain amount for the first bet, say ten dollars, and adding a dollar every time the bet is lost, or taking off a dollar every time a bet is won. If the player wins as many bets as he loses, and there is no percentage against him, he gets a dollar for every bet he wins, no matter how many bets he makes, or in what order the bets are won and lost, so that the number won equals the number lost. That this is so may be easily demonstrated by setting down on a sheet of paper any imaginary order of bets, such as the ten shown in the margin, five of which are won, and five lost; the net profit on the five bets won being five dollars. No matter how correctly the player may be guessing, and how much the luck runs his way, he wins smaller and smaller amounts, until at last he is “pinched off.” But if a long series of events goes against him his bets become larger and larger, but he must keep up the progression until he gets even. If ten bets go his way he wins $55; if ten go against him he loses $145.

It is said that Pettibone made a fortune playing progression at Faro, which is very likely, for among the thousands of men who play it the probabilities are that one will win all the time, just as the probabilities are that if a thousand men play ten games of Seven Up, some man will win all ten games. At the same time it is equally probable that some man will lose all ten.

Some players progress, but never pinch, keeping account on a piece of paper how many bets they are behind, and playing the maximum until they have won as many bets as they have lost. Against a perfectly fair game, with no percentage and no limit, and with capital enough to follow the system to the end, playing progression would pay a man about as much as he could make in any good business with the same capital and with half the worry; but as things really are in gambling houses and casinos, all martingales are a delusion and a snare. It is much better, if one must gamble, to trust to luck alone, and it is an old saying that the player without a system is seldom without a dollar. It is the men with systems who have to borrow a stake before they can begin to play.

Such matters as calculating the probability of a certain horse getting a place, the odds against all the horses at the post being given, would be out of place in a work of this kind; but those interested in such chances may find rules for ascertaining their probability in some of the following text books.

TEXT BOOKS.