The Mathematics and Misconceptions of Games of Chance
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All games of chance – whether casino games such as roulette, craps, blackjack and slots, or lottery and bingo, or card games such as poker or bridge – rely on certain basic statistical and probabilistic models. Uncertainty is built into them, which is what makes games ‘fun’ to play and also explains their continued existence. Casino games would never run if ‘the house’ wasn’t confident that they’d always win in the end. The mathematics of the games, including their rules and payout schedules, assures the house will profit in aggregate, regardless of individual behaviour.
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In mathematical terms, this guarantee is expressed through the fact that the house edge (HE) of a game is positive. The expected value of a bet (EV) is defined as follows: (probability of winning) × (payoff if you win) + (probability of losing) × (loss if you lose). The HE of a game is defined as the opposite of the expected value calculated for all possible bets (HE = −EV).
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For example, in European Roulette, a wheel spins and you have to predict where a small ball will land. There are 37 numbered slots (0 to 36). If you bet $1 on one numbered slot (called a straight-up bet), the payoff is 35 times what you bet, and the probability of winning is 1/37. So the EV of that bet is: (1/37) × $35 + (36/37) × (−$1). That is about −$0.027 or, as a percentage, 2.7 per cent of the initial bet. EV can be read as an average; in our example, you might expect to lose on average $2.70 at every 100 plays with that bet over the long run. This means that European Roulette has a house edge of 2.7 per cent. This is the house’s share of all the income produced by that game in the form of bets over the long run.
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From a player’s point of view, a positive house edge should mean that she can’t make a living off that game: over the long run, the house will have an advantage. That’s why a pragmatic principle of safe gambling behaviour is: ‘When you make a satisfactory win, take the money and get out of there.’Gamblers cultivate a whole spectrum of misconceptions that popularize games of chance. The ‘gambler’s fallacy’ is the belief that a series of bad plays will be followed by a winning outcome, in order for the randomness to be ‘restored’. The conjunction fallacy is the idea that the gambler estimates the probability of a combination of events to be higher than the probability of any one of those events. In other words, when someone uses addition rather than multiplication to estimate the probability of two or more mutually independent events. For example, in sports betting, someone might bet once on several ‘almost sure’ outcomes occurring together, thinking that it’s likely that all the house’s favourite teams will win – ignoring the fact that the product of the probabilities of several wins is a number significantly lower than the probability of any individual win. Another gambling black hole is the near-miss effect, when an outcome differs just a little from a winning one, which induces the gambler to believe that she was ‘so close’ that she should try again.
