Most roulette players have a vague sense that the house has an edge. Fewer know exactly how big it is. Almost nobody can tell you the precise probability of any given bet hitting or how to calculate the expected value of a combined play. This guide covers all of it.
There are three versions of roulette you are likely to encounter: European (single zero), American (double zero) and French (single zero with La Partage). The math is different on each. Knowing which wheel you are playing on is the single most important decision you make before placing a bet.
The Basic Probability Formula
Every probability in roulette comes from one formula: the number of pockets your bet covers divided by the total number of pockets on the wheel.
Probability Formula
P(win) = Numbers covered / Total pockets
European: Total pockets = 37 (1-36 plus one zero)
American: Total pockets = 38 (1-36 plus zero and double zero)
A bet on red covers 18 numbers. On a European wheel the probability is 18/37 = 0.4865 or 48.65%. On American it is 18/38 = 0.4737 or 47.37%. That difference of 1.28 percentage points is entirely due to the extra green pocket.
Complete Probability Table
| Bet | Covers | European P(win) | American P(win) | Payout |
|---|---|---|---|---|
| Straight up | 1 | 2.70% | 2.63% | 35:1 |
| Split | 2 | 5.41% | 5.26% | 17:1 |
| Street | 3 | 8.11% | 7.89% | 11:1 |
| Corner | 4 | 10.81% | 10.53% | 8:1 |
| Six line | 6 | 16.22% | 15.79% | 5:1 |
| Dozen / Column | 12 | 32.43% | 31.58% | 2:1 |
| Even money | 18 | 48.65% | 47.37% | 1:1 |
Expected Value
Expected value tells you how much a bet costs you on average per dollar wagered. The formula multiplies each possible outcome by its probability and adds them together.
For a $1 straight-up bet on European roulette:
EV = (35 x 1/37) + (-1 x 36/37) = 0.9459 + (-0.9730) = -0.0270
That means every dollar bet on a straight-up number costs you 2.70 cents in the long run. This calculation produces the same result for every bet on the European wheel: -$0.027 per dollar wagered. The payout ratios are designed to produce this exact number.
On an American wheel the same calculation yields -$0.0526 per dollar. Nearly double.
House Edge by Wheel Type
| Wheel Type | House Edge (Standard) | House Edge (Even Money with La Partage) |
|---|---|---|
| European (single zero) | 2.70% | N/A |
| French (single zero + La Partage) | 2.70% | 1.35% |
| American (double zero) | 5.26% | N/A |
The French wheel is the same physical wheel as the European. The difference is the La Partage rule: when zero hits, even-money bets lose only half. Some tables use En Prison instead where your bet is held for the next spin. Both rules cut the effective edge on even-money bets to 1.35%.
Probability of Hitting at Least Once
Players often want to know the odds of a number appearing at least once over a series of spins. The formula uses the complement method.
At Least Once Formula
P(at least once) = 1 - (1 - P(single spin))^n
Where n is the number of spins.
For a single number on a European wheel over 37 spins:
P = 1 - (36/37)^37 = 1 - 0.3623 = 0.6377 or about 63.8%
Even after 37 spins there is a 36.2% chance your number has not appeared. After 100 spins the probability of it appearing at least once rises to 93.4%. After 200 spins it is 99.6%. But there is still a 0.4% chance it has not shown up in 200 spins. That is roughly 1 in 250. In a casino running 40 tables at 35 spins per hour this happens multiple times every single day.
The Independence Problem
Every spin is independent. The wheel has no memory. This is the single most important concept in roulette probability and the one most frequently ignored.
If red has hit 10 times in a row the probability of red on the next spin is still 48.65% on a European wheel. It is not lower because black is "due." It is not higher because red is "hot." The ball does not know what happened on the previous spin. The dealer does not know. The wheel does not know. The number that appeared 10 spins ago has zero influence on what appears next.
This is not a philosophical position. It is a measurable physical fact. The mechanics of a fair roulette wheel produce outcomes that are statistically independent. Any strategy built on the assumption that past results influence future outcomes is built on a foundation that does not exist.
Combining Bets
When you place multiple bets on the same spin each bet is resolved independently against the house edge. Placing $5 on red and $5 on the third column does not change the house edge on either bet. Your total action is $10 and the expected loss is $10 x 0.027 = $0.27 per spin.
The only thing combining bets changes is your coverage of the layout and the shape of your outcome distribution. You might create a play that wins on 24 of 37 numbers but the payout structure will be adjusted so that the house retains exactly 2.70% of your total action.
There is no combination of bets that produces a positive expected value. The geometry of the payout table was designed specifically to prevent this. It is not an oversight. It is the architecture of the game.
What the Numbers Actually Tell You
Probability does not tell you what will happen on your next session. It tells you what happens across thousands of sessions. A player who bets $25 on red for 100 spins has an expected loss of $67.50 on a European wheel. Their actual result on any given night might be +$300 or -$400. Both are normal outcomes that fall well within expected variance.
The value of understanding the math is not prediction. It is calibration. When you know the true cost of every bet you can make informed decisions about bankroll size, session length and risk tolerance. You stop chasing impossible outcomes and start managing probable ones.
Watch the Odds Play Out in Real Time
Our simulator tracks actual vs expected probability across thousands of spins. See the math converge with your own data.
Open the SimulatorThe wheel does not owe you anything. But it does tell you exactly what it costs. The question is whether you are listening.