The Math Behind Casino Games

Everyone knows Alan Turing as the genius behind the cracking of the Enigma code that helped defeat the Nazis in the Second World War.

In fact, there’s even a superb movie about Turing – The Imitation Game, which has Benedict Cumberbatch as Turing in an Oscar-nominated performance. But news broke this week that before turning his attention to the war, Turing was busy trying to work out a system to win at roulette.

We won’t dwell too much on Turing’s work on roulette because, quite frankly, we don’t have the capacity to understand it. But it is no surprise that Turing was enchanted by the idea of beating the house in roulette – many mathematicians have tried to do so before. Some have used chaos theory, whereas others have used systems. Some have been successful, others not so much.


But it all boils down to one overarching fact – these games have been around for centuries, and there remains debate over whether they are flawed (mathematically speaking) in terms of their probabilities.

Heads or Tails is the simplest of games

To illustrate, let’s strip it all back and look at one of the simplest games, Heads or Tails. Despite the advances in technology, you can still play traditional Heads or Tails games for real money at online casinos. There are some extra betting options for the game at online casinos to make it a bit more exciting, but the fundamentals are the same.

Now, as most are aware, the outcome of a coin toss is 50/50. In the short term, you might win or lose. But, with the laws of probability, you will end up breaking even if you played for a long time. 50/50 is not going to be beneficial for a casino operator, however. So online games will be programmed using random number generators to put the odds – ever so slightly – in favour of the house.

Playtech’s Heads or Tails, for example, has an RTP of 97.6% (which is actually quite good for a virtual game), meaning you would theoretically win 48.8 games from every 50 played. But, as with all casino games, you must appreciate that those odds are worked out over a huge number of plays – millions of coin tosses, potentially.

Blackjack sparks debate over who has the advantage

But if it is easy to understand where the casino’s advantage comes from the coin toss, what about a relatively more complicated game like blackjack? The popular card game is perhaps even more enigmatic than roulette because nobody can agree whether the house actually has an advantage or not.


In classic blackjack, the house edge is accepted to be around 2%. That means you would be expected to win 49 from 50 hands. But there is a caveat here – that 2% refers to inexperienced players. Experienced players believe that the house edge might be less than 1%, whereas pro players might believe the advantage lies with them.

The problem is that there isn’t agreement on whether skill or blind chance is more important in blackjack. The casino’s advantage comes by way of the fact that the dealer does not have to play their cards if the player goes bust.

But, on the other hand, the player has options – double down, card counting, etc. – whereas the dealer must play their hand according to set rules.

Roulette has enchanted scientists

And what about roulette? We can work out the probabilities and house edge through the layout of the wheel. The standard European roulette wheel is numbered 1-36 (alternating Red and Black) and has one zero pocket (green).

The payout for guessing where the ball will land in a straight-up bet is 35 to 1 (you win 35 times your stake + your original bet is returned), which means the house edge is around 2.7%.


In short, the casino’s advantage in roulette is that extra zero pocket. However, not all roulette is the same. American roulette has two green pockets (zero and double zero), but payouts are the same, meaning the house edge is around 5.25%.

In some French roulette variants, there is a rule – La Partage – which awards half the bet back to the player if they have made an even/odd or red/black bet and the result is zero. This lowers the house edge to 1.35%.

So, if we see the clear reasons for the house edge in roulette – the number of pockets compared to the payouts – how and why have scientists tried to beat the system?

The example of using chaos theory to win has been much celebrated, but there is, of course, a flaw – you need to learn chaos theory, and that won’t happen overnight. Indeed, you might need years of study at a top university to pull it off. Even then, it might not work.

But for many, it continues to be alluring to try. All three games we have mentioned – heads or tails, blackjack, roulette – look so simple on paper. But the numbers and probability behind them are enigmatic. These games have been around for centuries, yet they still enchant both players and mathematicians today.

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