Roulette Odds and Probability Explained

Roulette Odds and ProbabilityThe game of roulette is believed to have been developed following an attempt to create a perpetual motion machine by 17th century French polymath Blaise Pascal.

That is fitting because in many ways the maths behind the game have a certain beauty that many people can appreciate.

Beginning with the trivial, some have pointed out that it is interesting that the numbers on a roulette wheel add up to 666, the supposed “number of the beast”.

Satanic links or not, a more interesting beauty can be seen in the interplay between the probabilities and odds of the various bets.

As with many games, both casino and otherwise, what seems so simple at first glance is actually a perfectly conceived game that maintains a flawless balance between the chances of the house and those of the player.

No Skill, No Luck, Just Maths

Maths Background Chalkboard

Let us assume we are talking about a standard European wheel with just a single zero. In this case, we are sorry to break it to anyone who feels they have a roulette strategy, system or staking plan that “works”, but what you do as the player makes not a single jot of difference to the long-term expected outcome. Luck affects things in the short term, which is why players can and do win, sometimes huge amounts. But in the end, it is true to say that nothing beats the cold, hard maths of the game.

The house edge, the advantage the casino holds over the player, which is the result of the nuanced and flawlessly conceived maths that underpin roulette, will not change. No matter what we punters do. You can bet 10 chips on eight numbers every third spin following consecutive reds, you can cover all numbers apart from your ex-girlfriend’s date of birth, or you can simply bet £10 on black every time and it makes no difference whatsoever to the end result you can expect in the long term.

This is because, for every single bet that it is possible to make playing roulette, the odds change perfectly in line with the chances of that outcome happening. What do we mean by this? Well, if you bet on a single number, you risk half the amount, to win half the amount, with half the chance of success, that a player betting the same amount on two separate numbers has. Put another way, whilst a bet on even numbers is far more likely to win than a bet on zero, the payout for the less likely outcome is increased in exact inverse proportion to its chances of winning.

Odds, Probabilities & House Edge

The table below shows the key information for all of the major bets available playing single zero roulette. As you can see, the house edge on all bets remains the same. This means that a player betting on lucky number seven 10,000 times at £10 a go can expect to lose exactly the same amount as a player betting on red 10,000 times, with £10 wagered each time.

In both instances the player will expect to lose on average just £2,700. That might sound like a lot but it is just 2.7% (the house edge on all bets on single-zero roulette) of the massive £100,000 total stake.

Bet Payout Odds Probability of Winning House Edge
Straight up – any single number (including zero 35/1 One in 37 2.70%
Split – any two adjoining numbers 17/1 One in 18.5 2.70%
Street – any three horizontal numbers 11/1 One in 12.33 2.70%
Corner – any four adjoining numbers (including 0,1,2,3) 8/1 One in 9.25 2.70%
Six line – six numbers from horizontal rows (two) 5/1 One in 6.17 2.70%
Any 12 numbers (various groupings, including 1-12 and 13-24) 2/1 One in 3.08 2.70%
Any 18 numbers (various groupings including red, odd or low numbers) Even money One in 2.06 2.70%

In addition, any combination of the bets above works in exactly the same way. So three separate six lines pays out the same amount, with the same chances of success as simply backing a colour, or alternatively, backing your 18 favourite individual numbers. The risk to reward ratio is the same on every bet in other words and no bet at roulette is any better or any worse than any other bet.

Variance

Risky Ralph and Steady EddieOne thing that does change is the variance. Variance, in simple terms, is how your results in the short term differ from the expected average in the long term. Imagine twins heading to the casino for a game of roulette. Steady Eddie bets £10 on low numbers (an even money bet with just under a one in two success rate expected) every spin for 37 spins. Risky Rafe instead opts to put £10 on 13 every time for the same 37 spins. This bet can be expected to win one in 37 spins, paying out at 35/1 when it does.

If the twins played an infinite number of spins, they will achieve the same results – a small loss of 2.7% of their bankroll. However, over 37 spins the chances are their results will differ greatly. Steady Eddie will almost certainly win on a few spins. Believe it or not, the longest ever consecutive run of one colour (which has the same probability as backing either high numbers or low numbers) is reported to be 32, so even with the worst luck imaginable we can expect Eddie not to lose all of his money.

Given the low numbers (1-18 inclusive) have an 18 in 37 chance of winning, on average you would expect him to win 18 times. Because each spin gives him an almost 50% chance of success, a huge deviation from that 18 is unlikely. What’s more, because the payout for each win is even money, his overall results in terms of profit or loss will not vary much either.

That means that, for example, if the twins visit the casino 1,000 times in their lifetime, using the same strategy, the vast majority of the time Eddie will probably win or lose less than £50, losing more often than he wins. Over their 1,000 visits, covering 37,000 spins in total, Risky Rafe’s overall outcome is likely to be similar, with both twins around 2.7% down. However, Rafe’s journey to that point will be rather different.

Backing a single number delivers huge variance because the payout is so much bigger and the chances of winning so much smaller. On average, Rafe should win once every visit and yield, like his brother, a small loss each time. Their financial outcomes should, in fact, be identical, with 18 £10 wins out of 37 at evens delivering a return of £360 and a loss of £10; which is exactly the same as one win in 37 at 35/1.

However, it doesn’t take too much bad luck for Rafe to win none of his spins. In fact, it is virtually guaranteed to happen many times over the 1,000 visits we have spoken about. On these visits, Rafe loses all £370 and it is Eddie who pays for the taxi home. As we have seen, this is a fate that Eddie himself is incredibly unlikely to experience, ever.

On the other hand though, Eddie is unlikely to win more than £20 or £30 at a visit. He is very unlikely to win more than £60 or £70 and almost guaranteed, in all those 1,000 visits, never to hit a profit of £250 or more. That would require him to win 31 out of 37 spins, highly unlikely with around a 48% chance of winning each time.

In contrast, all it takes for Rafe to land a big win is for him to win twice, rather than the expected once. Such an outcome would yield a profit of £350 for the night and over the period of their regular visits, Rafe is all-but-certain to have such nights. He may even have lucky sessions that deliver three wins, or perhaps four. Both would yield profits that are literally impossible for his brother to deliver (even if he won all 37 spins). The champagne and the taxi are on Rafe on these nights.

How To Improve Your Odds At Roulette

American Roulette

So, whilst you can change the variance of your results, depending on your preference for lots of small wins and losses, or more frequent losses but the occasional big win, you cannot change the odds or the house edge. Or can you?

Well, there is one thing you can do when it comes to roulette to improve your odds and whilst not ground breaking, it is very simple and 100% effective. You can pick the right game. First of all and most crucially, never, ever play American roulette or any variant that has more than one zero. The zero, or zeroes, create the casino’s advantage over the player. Doubling the zeroes almost doubles the house edge, taking it from 2.70% to 5.26% and means that over time you will lose almost twice as much.

The odds at which your bets get paid out stays the same but there is an extra number on the wheel where you lose. The other thing to look out for is French roulette, which may be called La Partage roulette. We look at La Partage and En Prison rules elsewhere on the site but in short, these roulette variations give you a second bite of the cherry should the ball land in zero, cutting the house edge to less than 1.5%.