0 returned1 unit lost
The mathematics behind the wheel
RouletteOdds & RTP
Under standard roulette rules, each bet has a winning probability and a fixed payout. Together, they determine its average return — and the advantage built into the rules.
Understand the return37equally likely pockets
18 red · 18 black · one zero
97.30%theoretical RTP
standard European roulette
01 / The chance of winning
One wheel.
37 possibilities.
A fair European wheel has 37 equally likely pockets: 0–36. Your bet wins if the ball lands on a number it covers.
Straight-up 17
1/ 37
2.70% chance of winning
17 wins. The other 36 outcomes lose.
Red is not a 50/50 wager: it covers 18 outcomes and loses on the other 19, including zero. A dozen covers twelve numbers, giving a 12/37, or 32.43%, chance of winning.
02 / Probability meets payout
Different bets.
The same return.
A winning straight-up bet returns 36 units on a 1-unit stake. Red returns two. The lower payout balances the greater chance of winning.
Straight-up 171 winning outcome out of 37
36units returned on a win
35:1 net payout
0123456789101112131415161718192021222324252627282930313233343536
Only 17 wins, returning 36 units including the stake. The other 36 outcomes return nothing.
Possible return units36
1 / 37×36=36 / 37
Theoretical RTP97.30%
For one wager, the expected return weights each of the 37 outcomes equally. In 37 actual spins, numbers can repeat and others may not appear.
03 / The cost of zero
One extra loss.
No extra payout.
A Red wager wins on 18 outcomes. It loses on 18 black numbers and zero. A win still returns only two units.
18winning outcomes
2 units returned
19losing outcomes
nothing returned
Expected return(18 × 2 + 19 × 0) / 3797.30%
House edge1 − 36/37 = 1/372.70%
The house edge is the expected share of stakes retained by the casino. It is an average, not a fixed deduction from every bet.
RTP means Return to Player: the expected percentage of total stakes returned under the specified rules. A single Red bet returns two units or nothing, never a guaranteed 97.30% refund.
04 / The rules matter
What changes
when zero multiplies?
Ordinary winning payouts stay the same on wheels with more zeros. La Partage takes a different approach: the wheel stays the same, but eligible even-money bets lose only half their stake on zero.
Numbered pockets36
18 red · 18 black
Green zeros1
000000
Equally likely outcomes37
36 + 1 = 37
Expected return36 / 37
Theoretical RTP97.30%
House edge2.70%
Single zero with standard payouts. There is no half-stake refund on zero.
European reference 36/37 · 97.30% RTPOutcome counts, independent of physical wheel order.
1
A 1-unit bet on RedZero lands.
0.5 returned0.5 unit lost
The wheel still has 37 outcomes. Only the treatment of zero changes.
(18 × 2 + 0.5) / 37 ≈ 98.65% RTP for eligible even-money bets.
La Partage applies to even-money bets.
French roulette uses a single-zero wheel. Where La Partage is offered, the 98.65% RTP applies to eligible Red/Black, Odd/Even and Low/High bets. It does not apply to inside bets, dozens or columns, which retain the standard 2.70% house edge.
With En Prison, an eligible even-money stake stays on the table after zero. On a later non-zero result, a win returns the stake without profit and a loss forfeits it. The rules for repeated zeros vary and can change the RTP.
The American five-number exception
000123
Five winning outcomes out of 38 pay 6:1 net, returning seven units on a winning one-unit stake. RTP is 35/38 ≈ 92.11%; house edge is 3/38 ≈ 7.89%, above the ordinary American rate.
05 / The amount wagered
Every stake counts.
Expected loss is total wagered multiplied by the house edge. Twenty 5-unit bets total 100 units wagered, regardless of your starting bankroll. Winnings you bet again count too.
These are mathematical averages, not a prediction of a session. Actual losses can be smaller or larger, and a session can finish in profit. La Partage figures apply only to eligible even-money wagers.
06 / Expected and observed
Ten spins.
Three different returns.
Actual RTP is total returned divided by total staked. With one unit on Red each spin, four wins in ten return eight units: 80% actual RTP. Five wins give 100%; six give 120%.
Total staked1010 bets × 1 unit
Total returned84 wins × 2 units
Actual RTP80%returned ÷ staked × 100
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3
4
5
6
7
8
10
11
Theoretical RTP97.30%unchanged by the session result
A single Red win produces 200% actual RTP. A straight-up win produces 3,600%. Neither changes the long-run expectation. Even 100, 1,000 or 10,000 spins cannot guarantee a particular return; earlier losses do not make future wins due.
07 / The shape of a result
How often.
How much.
Red wins more often and pays less. Straight-up 17 wins rarely and pays much more. Both retain the same 97.30% theoretical RTP.
One unit stakedLosing outcomesWinning outcomes
Red
Frequent small wins
19/ 37
Net result−1unit
18/ 37
Net result+1unit
Straight-up 17
Rare large wins
36/ 37
Net result−1unit
1/ 37
Net result+35units
Each dot is one of the 37 possible outcomes. The signed amounts are net profit or loss on a 1-unit wager.
Both bets have the same expected net result−1 / 37 unit per unit staked
97.30% theoretical RTP · 2.70% house edge
08 / Keep the figures straight
Odds & payouts.
Net payout is profit on a win. The original stake is returned as well.
Standard European bets
97.30% RTP
2.70% house edge
36/37 returned · 1/37 retained in expectationInside bets
| Bet | Covered | Win chance | Net payout |
|---|---|---|---|
| Straight-up | 1/37 | 2.70% | 35:1 |
| Split | 2/37 | 5.41% | 17:1 |
| Street | 3/37 | 8.11% | 11:1 |
| Corner | 4/37 | 10.81% | 8:1 |
| Six Line | 6/37 | 16.22% | 5:1 |
Outside bets
| Bet | Covered | Win chance | Net payout |
|---|---|---|---|
| Dozen / Column | 12/37 | 32.43% | 2:1 |
| Red / BlackOdd / EvenLow / High | 18/37 | 48.65% | 1:1 |
Rules & variants
| Rules | Pockets | RTP | House edge |
|---|---|---|---|
| European | 37 | 97.30% | 2.70% |
| French · La PartageEligible even-money bets only | 37 | 98.65% | 1.35% |
| AmericanOrdinary bets | 38 | 94.74% | 5.26% |
| Triple ZeroConventional payouts | 39 | 92.31% | 7.69% |
The American five-number bet is separate: 5/38 chance, 6:1 net payout, 92.11% RTP and 7.89% house edge.
09 / Questions about the odds
Are payout odds the same as odds against winning?
No. A straight-up wager has one winning outcome and 36 losing outcomes, so the odds against it winning are 36:1. The standard net payout is 35:1. The gap produces the house edge.
Can a betting system change the RTP?
Changing standard bets or stake sizes does not remove the house edge. Increasing stakes after a loss changes how much money is at risk and how quickly a bankroll can run out. It does not change the chance of winning the next spin.
Read about strategies and their limitsDoes a larger sample guarantee the theoretical RTP?
No. Even a large sample can finish above or below the theoretical RTP. Expected loss is an average; actual loss is total wagered minus total returned. Earlier losses do not make a future win more likely.
Standard European roulette
Different bets.
The same 36/37.
Covering more numbers increases the chance of a win and reduces its payout. Under standard European rules, the theoretical return stays at 97.30%.