Play with the numerically innocent. Admittedly, this is not much fun. Most of what follows consists of instructions for the moral equivalent of stealing from babies.
From here on in this entry, the winner will be referred to as the 'player' and losers will be referred to by the more flattering name of 'benefactors'. Whatever the size of the bet, we will refer to it as one unit.
The first thing to notice is that the size of the unit has no effect on the event whose outcome is the subject of your bet; this means that there is no system of betting that can improve your chances of winning by varying the size of the wager. Only one thing counts:
Getting on the right side of the figures
This is not difficult. In professional terms, it means 'own the casino'. In private life there are many choices of game, from the blindingly simple to the blindingly difficult, from which to take your pick.
A simple dice game
Wager repeatedly on the outcome of a single throw of two dice. Offer even money to your benfactor, on the outcome totalling 2, 3, 4, 10, 11, 12 on the one hand, or 5, 6, 7, 8 or 9 on the other. Point out that you will take on the five numbers and he can take the six. You win: see the figures below for details.
You are practically forced to win, as the odds are 2:1 that a pair of dice will show one of the five middle totals rather than one of the six outer ones. If the reader now feels the threat of 'lies, damned lies and statistics' about to be applied, the reader has a point.
But if you do not trust your grasp of mathematics, you should promise yourself never to accept bets concerning anything in which you are not an expert. You also owe it to yourself to avoid large lotteries; you should feel better for having given a tiny sum to a cause or an individual of your choice rather than to a randomly-created millionaire who is almost certain to be nobody you know or approve of. Alternatively, you may choose to accept the role of benefactor to the many charming people eager to hazard their money with you. Many rich people do this; it is their chosen form of entertainment, or, who knows, of charity.
Simple figures (essential)
Two to one may seem a surprising weight in favour of the five middle numbers, but it is easily demonstrated.
One die has six possible outcomes, for each of which the second die has six, totalling thirty-six possibilities when two dice are thrown. Of the thirty-six possible outcomes, twelve amount to 2, 3, 4, 10, 11 or 12, while twenty-four amount to totals between five and nine; Q.E.D.
The odds are so heavily in your favour that you can concede any other number to your benefactor (perhaps one that has been turning up frequently in your favour; benefactors tend to be moved by such observations) and still count on winning, so long as it's not 7. There are eighteen ways to score 2, 3, 4, 7, 10, 11 or 12, and eighteen ways to score 5, 6, 8 or 9; in such a bet you forfeit the name of 'player' and become an ordinary gambler.
Technical figures (optional)
Imagine a game between Bob and Alice. Alice takes the five middle numbers, Bob the six outer ones. The probability of Alice winning a game is 2/3.
For Alice to show a net profit she has to win more than half the games.
Let's call the number of games Alice wins X, and suppose that they play n games. Then
The probabilty of Alice not making any money, i.e. losing half or more of the games, is given by:
Here's a table showing some of the probabilites for varying n
|n||P(X ≤ n/2)|
That is to say, after only twenty throws, Alice has a more than ninety percent likelihood of being ahead; and by the time forty throws have been made, that has risen to almost 98%.
An even simpler card game
Take any two cards from a pack and, before revealing them, bet with two benefactors that they will be of different colours. If they are both black, one benefactor wins, if they are both red, the other wins. The player keeps the third option, that they are one black and one red. Benefactors are not to notice that there are not in fact three possible outcomes; there are four, two of which favour the player.
Admittedly your chances are only fifty-fifty, but this means that in half of the transactions you are paying out one unit and in the other half you are receiving two units; which is nice.
This can equally well be played by flipping two coins or betting on equally-likely pairs of any sort. It is mentioned in the context of cards for the following historical reason. One Researcher's father played this game in the nineteen-thirties, arising from an advertising gimmick: a cigarette manufacturer gave away two random playing cards with each packet of fags1, and the player began betting pennies with his fellow-workers whenever one of them opened a packet. He felt no remorse for this blatant robbery, since the business they were in was accountancy, and they should have known better.
One can earn a living very comfortably on much smaller odds than 2:1. The casino's profit in roulette is made by adding a thirty-seventh option to the eighteen black and eighteen red numbers on the wheel: the inconspicuous green zero. This addition tilts the odds of the house winning when the benefactor bets on 'fifty-fifty' chances such as 'red', 'black', 'odd' or 'even'2 from 18:18 (1 to 1, or even money) to 19:18 (just over 1.05 to 1), which is a sufficient difference to pay for the rent, furninshings and staff of a luxurious casino, and provide profits for its owner, so long as sufficient benefactors continue to turn up. Roulette offers some of the best, that is least unfavourable, odds to the gambler3; most other casino games favour the house more.
The best casino gambling strategy therefore is to decide how much you want to risk in an evening, put it all on red (or, if you prefer, black) in one bet at roulette, and leave, with your winnings or without your investment. Agreed, this is not much fun.
In real life, your best bet is to choose a game of skill that you like, study it until you are better than average, and take on the world. This can be fun, whether your game is backgammon, chess or the stock market. Remember though that maths is the word of God, and never transgress against the figures.