Mathematics

# How Probability is used in Games of Chance

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Human beings seem to have a natural instinct for gaming and gambling. The Egyptians were playing a board game called Senet over 5000 years ago, and it’s likely that simpler games preceded this. Our minds seem to be programmed to want to play, and to want to win. But something we aren’t so good at is evaluating the probabilities affecting our games, and applying those in a way that maximises our chances of vanquishing the opposition.

It is important to distinguish between games of pure chance, games of pure skill, and the full range of variations between these two extremes. On the ‘pure chance’ end of the scale are games like snakes-and-ladders, or roulette, where all the player can do is wait for physics to do its thing and then climb that ladder or rake those winnings off red nineteen. Provided the dice or table are fair, there is nothing the player can do to improve their chances of winning - unless their personal deity pays attention to prayer for profit!

‘Pure skill’ is exemplified by chess. Perhaps there is an element of chance in whichever player gets the white pieces, but after that the entire game is dictated by the skills of the players. Even sports such as soccer and tennis have more of chance the element than chess - you can never trust the vagaries of the weather and the referee’s eyesight!

Between these two extremes we find all those games where there is some kind of randomising influence, be it dice, the shuffling of a deck of cards, or even a decision made by an unbiased non-participant. Backgammon and poker are good examples. Both games rely heavily on elements of chance, but involve enough personal choice that a skilled player can come out on top in the long term.

So how do we need to calculate and evaluate probabilities and how should our game playing change based on this information? It’s simplest to begin with games of pure chance...

There are a couple of considerations to make. Firstly, are you simply playing for fun, or are you playing with the intention of profiting from the game? If it is the former - perhaps you’re playing the aforementioned snakes-and-ladders against your kids - then don’t worry about it! Simply enjoy the game for the social experience; groan when the dice go against you and laugh when lady luck finally blows you a kiss.

However, if you’re looking to gain something from the game, presumably you’re also risking something of value. Understanding how the game works, calculating the probabilities and then deciding whether the game is acceptably ‘fair’ is important. In a completely fair game of chance, the odds of winning should perfectly match the reward for winning. If you’re betting against a friend on the flip of a coin, then it would be fair to bet a dollar to win a dollar. On the roll of a fair die, you should bet a dollar to win six. In the long term both players will come out even.

In an unfair game, one player has to pay more to play the game than is justified by the odds. There isn’t a single game you can play in the casino that is fair. They are all set up so that for every dollar you gamble, you’ve only got a 49% chance of winning a dollar - so in the long term they win and you lose. Of course, everyone is aware of this and still plays. Why? Because visiting the casino is not merely about winning money. It’s about the thrill of the possibility of winning big, and it’s about having a great night out.

How about the lottery? Lotteries are generally grossly unfair. However, when people buy a lottery ticket, it’s not with the thought of long term profit (lotteries vary, but most players worldwide will end up with 30-50% of their ‘investment’) but with that 14-million-to-one chance of ending up rich beyond the dreams of avarice.

So, when it comes to games of pure chance, just work out how much you have to pay to play, work out what your expected return is, and if you don’t mind losing the difference, and you can cope with losing the lot, then have fun! The good thing about games of pure chance is that you can calculate all the odds beforehand, and provided the rules don’t change, you always know exactly where you stand.

How about more complex games? To some people, poker is essentially a game of chance, and they view it the same way they view craps or baccarat. You get dealt the cards that fate has handed you, and whoever wins wins. In a two-horse-race, one player will have a better hand than the other, and it’s a 50:50 chance who that person is - just like flipping a coin. The complexity arises from the constantly changing probabilities as the game is played, and the ability of the players to make choices about how much to risk at any moment of the game.

If you have an incredibly analytical mind, and provided your opponents are unable to gain any information about your hand from your behaviour, then poker can be approached like any other pure chance bet. Work out the odds of winning, and then make sure that the expected return on your bet is better than the risk. If you think your two aces have a 30% chance of winning, then it’s well worth betting five dollars for a chance at a twenty dollar pot. If it’s a seventeen dollar pot then it’s barely worth it, and any less should have you throwing your cards in. Of course, calculating exact figures like this in the heat of battle isn’t easy! Good players win not by spotting ‘tells’ but by having a better understanding of the risk and reward as they play the game.

So, in games determined by a chance event, but which rely on personal decision-making regarding risk, it’s up to the player to make choices that maximise their chances. The problem with this, and this takes us back to the beginning of the article, is that human beings are pretty terrible at evaluating probabilities and risk. Not only do we get that ‘lucky feeling’ which tempts us into poor risks, but our brains simply cannot cope with the probabilities in unusual games. I  thoroughly recommend that anyone interested in the counter-intuitive nature of probability looks at the Monty Hall problem, and Simpson’s paradox, which has a bearing on sporting statistics.

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