Roulette Mathematics
If you want to excel at roulette, then you really need to understand the maths behind it. True odds are the real world chance of a certain outcome, whereas the house odds are the payout ratio for the same scenario and are typically slightly lower. Being aware of and understanding the odds is essential to becoming a successful player and the best thing is it’s relatively simple, but also quite beautiful. I will do my best to outline the mathematics behind this game in simple terms, but without missing any important details.
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The House Edge
Firstly, we consider a European roulette table, so that there are thirty-six numbers which are red and black, and a green zero. Placing an outside bet (e.g on red) gives you an 18/37 chance of winning, similarly for black. That means there is a 1/37 (=0.027) chance of landing on green, in which case the casino keeps the bets. The payout for winning a bet on red is 1:1 but this is not reflected in the odds. Like we mentioned above there is an 18 in 37 chance for us to win this bet, meaning that in 19 out of 37 we would lose.
In order to calculate the house edge, we will apply the following formula: (Odds Against Winning – House Odds) * Probability of Winning * 100 = house edge percentage. Therefore if we take the formula and apply it, we will have the following equation: (19/18 – 1/1) * 18/37 * 100 = 0.027027027. Therefore, we come to the conclusion that the casino has an edge of 2.7% at European Roulette. The difference between the winning and losing scenarios is 1 in 37, which constitutes the house edge. This is also the case for French Roulette, which uses the same roulette wheel, payout ratios, and therefore the same odds and house edge.
Similarly, we can calculate the house edge for American Roulette, which uses a 38-pocket wheel. In this variation of roulette, there are 36 black and red pockets, as well as two green ones for the zero and the double zero. This would normally not be an issue, but American Roulette uses the same payout ratios as European and French without adjusting them for the addition pocket, which complicates things for players. For example, the chance to win a Straight Up bet (single number) is 37:1, but the payout for that is 35:1. Here we notice the larger difference in true and house odds. If we take the formula and apply in this case we get (37/1 – 35/1) * 1/38 = 2/1 * 1/38 = 2/38 = 0.0526315. In terms of percentages (0.0526315*100), we come to the conclusion that the house edge for American Roulette is ~5.26%.
There is a bet exclusive to American Roulette, which is not featured in the other versions. This is the so-called Basket Bet, which is a bet on a cluster of five adjacent numbers, more specifically, on 0, 00, 1, 2, 3. What is special about this wager is that it comes with a higher house edge than the others, earning the name Sucker Bet among some gambling circles. Let’s examine why this is. For starters, this is a bet made with one chip on all five numbers and it pays 6:1. When we apply the formula we will come to the following result – (33/5 – 30/5)*5/38 = 0.07894736842, or ~7.89%. As you can clearly see, this bet comes with a very harsh house edge when compared to the other wagers. The Basket Bet is the only of its kind to deviate from the norm and feature a different house advantage, making it both unique and unfavourable at the same time.
In short, we have come to the following conclusions:
- For European roulette, the house edge is 2.7%, due to the difference between true odds and casino odds.
- For American roulette, the house edge is higher at 5.26%, due to the game featuring an additional zero pocket, while failing to account for that in the paytable.
Roulette Bets
There are numerous different bets that you can place while playing roulette and each comes with its own specific name and payout. In this point, we will go over all the available wagers that you can place at the roulette table and go into detail for the important ones. Generally, there are two types of bets in this game – Inside and Outside bets.
Inside Bets
The Inside Bets refer to bets made inside the number grid on European and American Roulette. These wagers are high risk, high reward and if you manage to win several of them in a row, you might leave the table significantly richer than before. Below you will find all Inside Bets and their payouts and probabilities.
- Straight Up – This is a bet placed with a single chip on a single number in the grid. The chip is placed directly on the number. This is the highest paying wager in the whole game, paying 35:1 for every successful bet, while the chance of winning is 2.7% (1 in 37) and 2.63% (1 in 38) in European and American Roulette, respectively.
- Split – This is a bet on two neighbouring numbers using a single chip, placed on the line separating the two numbers. The payout for a winning Split bet is 17:1 in both versions of the game. The chance of winning this bet is 5.41% (2 in 37) and 5.26% (2 in 38) for European and American Roulette, respectively.
- Street – This is a bet on three adjacent numbers in a line. The chip is placed on the edge of the line and the payout for this bet is 11:1. The chance to win this wager is 8.11% (3 in 37) and 7.89% (3 in 38) for European and American Roulette, respectively.
- Square – This is a bet on 4 neighbouring numbers simultaneously using a single chip, which is placed on the border of the four numbers. A winning Square bet pays 8:1 and your chances of winning are 10.81% (4 in 37) and 10.53% (4 in 38) for European and American Roulette, respectively.
- Basket – The Basket Bet is one that we have covered above. It is only present in American Roulette and it involves the numbers 0-00-1-2-3. The payout for this bet is 6:1, while the chance of winning is 13.16% (5 in 38). Worth noting is that this bet comes with a higher house edge than any other wager in the game at 7.89%. For this reason, it is generally a good idea to avoid this wager in favour of others.
- Line – This is a bet on six numbers using a single chip. This bet can also be referred to as a Double Street since it involves two neighbouring lines. The chance to win this wager is 16.22% (6 in 37) and 15.79% (6 in 38) for European and American Roulette, respectively. The payout for a Line bet is 5:1.
So how likely is a number to come up? Well, this is obvious for a single inside bet. The probability of any particular number coming up is 1/37 = 2.7%. But how likely is that number to come up again? The answer is (1/37) x (1/37) = 0.73%. The chances are very slim, and they quickly decrease. In fact, the probability for a number to come up n-times in a row is given by (1/37)^n.
For this reason, many players stick to outside bets, as while the payout is much smaller, the chances of making a return on your bet are far greater, meaning players can gamble for longer and seemingly lose less money.
Outside Bets
Outside Bets are slightly different than their Inside counterparts, primarily due to the fact that these are bets placed on larger groups of numbers. Whereas the largest Inside Bet includes 6 numbers, the smallest Outside one involves 12. Below you will find all the Outside Bets, as well as, their payouts and probabilities.
- Red or Black – This is a bet on whether the ball will land in a particularly coloured pocket. The payout for this bet is 1:1, while the chances of winning it are 48.65% (18 in 37) and 47.37% (18 in 38) for European and American Roulette, respectively.
- Odd or Even – This is a bet on whether the ball will land on an even number or an odd one. The chances of winning this bet are 48.65% (18 in 37) and 47.37% (18 in 38) for European and American Roulette, respectively, while the payout is 1:1.
- Low or High – This is a bet on the winning number will be in two range of numbers. The two possibilities are between 1 – 18 or between 19 – 36. The payout for this bet is 1:1, while the chances of winning are 48.65% (18 in 37) and 47.37% (18 in 38) for European and American Roulette, respectively.
- Column or Dozens – These two bets are practically identical, with one having you bet on one of the three columns, while the other is a bet on a range of numbers (1-12, 13-24, 25-36). Both pay 2:1, but the odds of winning are only 32.43% and 31.58% for European and American Roulette, respectively.
You may be thinking, “what if I bet on two columns, then I have a $64.8% chance of winning”. This is correct, but give it a thought. Let’s say you place a bet of £10 on each column. Each win will give you £20 profit, but you will lose £10 on the other bet so that’s only a £10 profit in total. Any number that comes up that isn’t covered by your bet will result in a loss of £20.
So if we look at the bigger picture, over 1000 bets we have:
648 wins – Profit: 648 x 10 = £6480
352 losses – Profit: 352 x -20 = -£7040
Total profit: -£560
In the long run, the casino still has the advantage, it’s just a little trickier to see how.
Of course, the mathematics behind the bets is essentially the same for all types of bets. Below is a table of the odds for each type of bet (for European roulette) and the corresponding payout.
| Bet | Payout | European Roulette Odds | EU Roulette House Edge | American Roulette Odds | American Roulette House Edge |
|---|---|---|---|---|---|
| Straight Up | 35:1 | 2.70% | 2.70% | 2.63% | 5.26% |
| Split | 17:1 | 5.41% | 2.70% | 5.26% | 5.26% |
| Street | 11:1 | 8.11% | 2.70% | 7.89% | 5.26% |
| Square | 8:1 | 10.81% | 2.70% | 10.53% | 5.26% |
| Basket | 6:1 | x | x | 13.16% | 7.89% |
| Line | 5:1 | 16.22% | 2.70% | 15.79% | 5.26% |
| Columns | 2:1 | 32.43% | 2.70% | 31.58% | 5.26% |
| Dozens | 2:1 | 32.43% | 2.70% | 31.58% | 5.26% |
| Low or High | 1:1 | 48.65% | 2.70% | 47.37% | 5.26% |
| Even or Odd | 1:1 | 48.65% | 2.70% | 47.37% | 5.26% |
| Black or Red | 1:1 | 48.65% | 2.70% | 47.37% | 5.26% |
How much doe you made doin this lad?
Hello I think roulette predicted with math
An insightful mathematical analysis of roulette betting systems is in Chapter 9 of my book Surpries in Probability- Seventeen Short Stories
Since all roulette bets, except for the basket bet, have the same house edge, why doesn’t a split bet pay 35/2:1?
Doesn’t that make 35:1 the best bet?