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Exact casino math

Roulette

Deterministic wheel math for American and European layouts, including the American first-five exception.

Ship exact per-bet probability, expected value, and house-edge tables for every common inside and outside bet.

Rules focusThe rule assumptions and model boundary this page uses for its calculations.
Standard table bets are modeled as covered pockets plus a payout multiplier. Wheel size does the rest.
OutputsThe main numbers or result groups this page will produce after you run the analysis.
Win probabilityLoss probabilityEV per unitHouse edge

Build a full layout bet, edit individual stake sizes, and see the combined positive-outcome rate instead of a single generic bet family.

Rules

Wheel math

Every roulette wager is just a set of covered pockets and a posted payout. The wheel size determines the rest.

Standard bets sit on a 37-pocket wheel, dropping the normal house edge to 2.70%.

  • European wheels use 37 pockets and keep the standard layout payouts.
  • American wheels add 00, which raises the base edge on most wagers to 5.26%.
  • The American 0-00-1-2-3 top line remains a separate outlier with a worse edge than the rest of the layout.

Reference

Single-bet odds table

This stays as the quick reference view. The analyzer below is where multiple simultaneous bets get merged into one exact profile.

How to read EV and net resultEV per unit is the average profit or loss on a 1-unit bet over many spins. In the merged layout analyzer below, net result means the realized profit on a specific landing pocket after subtracting every losing chip and paying each winning bet at payout-to-1 odds.
BetPayoutWin rateEV / unitHouse edge
Straight upA single number.35:12.70%-0.027u2.70%
SplitTwo adjacent numbers.17:15.41%-0.027u2.70%
StreetThree numbers in a row.11:18.11%-0.027u2.70%
CornerFour numbers sharing a corner.8:110.81%-0.027u2.70%
Six lineTwo adjacent streets.5:116.22%-0.027u2.70%
Dozen / columnAny 12-number outside grouping.2:132.43%-0.027u2.70%
Even-money outsideRed/black, odd/even, or high/low.1:148.65%-0.027u2.70%

Analyzer

Build a full roulette layout wager

Use the printed board itself to place chips on straight-ups, splits, corners, streets, six-lines, and outside bets, then fine-tune stake sizes in the active list.

Click directly on the printed board to drop a 1u chip. Hover reveals the active hit area, right click removes a chip, and exact stake sizes can still be edited in the active bets list below.

European Single Zero Layout1234567891011121314151617181920212223242526272829303132333435360SINGLE ZEROEuropean board1st 122nd 123rd 122 TO 12 TO 12 TO 11 TO 18EVENREDBLACKODD19 TO 360 straight up: left click adds 1u, right click removes 1u1 straight up: left click adds 1u, right click removes 1u2 straight up: left click adds 1u, right click removes 1u3 straight up: left click adds 1u, right click removes 1uSplit 1-2: left click adds 1u, right click removes 1uSplit 2-3: left click adds 1u, right click removes 1uStreet 1-3: left click adds 1u, right click removes 1uSplit 1-4: left click adds 1u, right click removes 1uSplit 2-5: left click adds 1u, right click removes 1uSplit 3-6: left click adds 1u, right click removes 1uCorner 1-2-4-5: left click adds 1u, right click removes 1uCorner 2-3-5-6: left click adds 1u, right click removes 1uSix line 1-6: left click adds 1u, right click removes 1u4 straight up: left click adds 1u, right click removes 1u5 straight up: left click adds 1u, right click removes 1u6 straight up: left click adds 1u, right click removes 1uSplit 4-5: left click adds 1u, right click removes 1uSplit 5-6: left click adds 1u, right click removes 1uStreet 4-6: left click adds 1u, right click removes 1uSplit 4-7: left click adds 1u, right click removes 1uSplit 5-8: left click adds 1u, right click removes 1uSplit 6-9: left click adds 1u, right click removes 1uCorner 4-5-7-8: left click adds 1u, right click removes 1uCorner 5-6-8-9: left click adds 1u, right click removes 1uSix line 4-9: left click adds 1u, right click removes 1u7 straight up: left click adds 1u, right click removes 1u8 straight up: left click adds 1u, right click removes 1u9 straight up: left click adds 1u, right click removes 1uSplit 7-8: left click adds 1u, right click removes 1uSplit 8-9: left click adds 1u, right click removes 1uStreet 7-9: left click adds 1u, right click removes 1uSplit 7-10: left click adds 1u, right click removes 1uSplit 8-11: left click adds 1u, right click removes 1uSplit 9-12: left click adds 1u, right click removes 1uCorner 7-8-10-11: left click adds 1u, right click removes 1uCorner 8-9-11-12: left click adds 1u, right click removes 1uSix line 7-12: left click adds 1u, right click removes 1u10 straight up: left click adds 1u, right click removes 1u11 straight up: left click adds 1u, right click removes 1u12 straight up: left click adds 1u, right click removes 1uSplit 10-11: left click adds 1u, right click removes 1uSplit 11-12: left click adds 1u, right click removes 1uStreet 10-12: left click adds 1u, right click removes 1uSplit 10-13: left click adds 1u, right click removes 1uSplit 11-14: left click adds 1u, right click removes 1uSplit 12-15: left click adds 1u, right click removes 1uCorner 10-11-13-14: left click adds 1u, right click removes 1uCorner 11-12-14-15: left click adds 1u, right click removes 1uSix line 10-15: left click adds 1u, right click removes 1u13 straight up: left click adds 1u, right click removes 1u14 straight up: left click adds 1u, right click removes 1u15 straight up: left click adds 1u, right click removes 1uSplit 13-14: left click adds 1u, right click removes 1uSplit 14-15: left click adds 1u, right click removes 1uStreet 13-15: left click adds 1u, right click removes 1uSplit 13-16: left click adds 1u, right click removes 1uSplit 14-17: left click adds 1u, right click removes 1uSplit 15-18: left click adds 1u, right click removes 1uCorner 13-14-16-17: left click adds 1u, right click removes 1uCorner 14-15-17-18: left click adds 1u, right click removes 1uSix line 13-18: left click adds 1u, right click removes 1u16 straight up: left click adds 1u, right click removes 1u17 straight up: left click adds 1u, right click removes 1u18 straight up: left click adds 1u, right click removes 1uSplit 16-17: left click adds 1u, right click removes 1uSplit 17-18: left click adds 1u, right click removes 1uStreet 16-18: left click adds 1u, right click removes 1uSplit 16-19: left click adds 1u, right click removes 1uSplit 17-20: left click adds 1u, right click removes 1uSplit 18-21: left click adds 1u, right click removes 1uCorner 16-17-19-20: left click adds 1u, right click removes 1uCorner 17-18-20-21: left click adds 1u, right click removes 1uSix line 16-21: left click adds 1u, right click removes 1u19 straight up: left click adds 1u, right click removes 1u20 straight up: left click adds 1u, right click removes 1u21 straight up: left click adds 1u, right click removes 1uSplit 19-20: left click adds 1u, right click removes 1uSplit 20-21: left click adds 1u, right click removes 1uStreet 19-21: left click adds 1u, right click removes 1uSplit 19-22: left click adds 1u, right click removes 1uSplit 20-23: left click adds 1u, right click removes 1uSplit 21-24: left click adds 1u, right click removes 1uCorner 19-20-22-23: left click adds 1u, right click removes 1uCorner 20-21-23-24: left click adds 1u, right click removes 1uSix line 19-24: left click adds 1u, right click removes 1u22 straight up: left click adds 1u, right click removes 1u23 straight up: left click adds 1u, right click removes 1u24 straight up: left click adds 1u, right click removes 1uSplit 22-23: left click adds 1u, right click removes 1uSplit 23-24: left click adds 1u, right click removes 1uStreet 22-24: left click adds 1u, right click removes 1uSplit 22-25: left click adds 1u, right click removes 1uSplit 23-26: left click adds 1u, right click removes 1uSplit 24-27: left click adds 1u, right click removes 1uCorner 22-23-25-26: left click adds 1u, right click removes 1uCorner 23-24-26-27: left click adds 1u, right click removes 1uSix line 22-27: left click adds 1u, right click removes 1u25 straight up: left click adds 1u, right click removes 1u26 straight up: left click adds 1u, right click removes 1u27 straight up: left click adds 1u, right click removes 1uSplit 25-26: left click adds 1u, right click removes 1uSplit 26-27: left click adds 1u, right click removes 1uStreet 25-27: left click adds 1u, right click removes 1uSplit 25-28: left click adds 1u, right click removes 1uSplit 26-29: left click adds 1u, right click removes 1uSplit 27-30: left click adds 1u, right click removes 1uCorner 25-26-28-29: left click adds 1u, right click removes 1uCorner 26-27-29-30: left click adds 1u, right click removes 1uSix line 25-30: left click adds 1u, right click removes 1u28 straight up: left click adds 1u, right click removes 1u29 straight up: left click adds 1u, right click removes 1u30 straight up: left click adds 1u, right click removes 1uSplit 28-29: left click adds 1u, right click removes 1uSplit 29-30: left click adds 1u, right click removes 1uStreet 28-30: left click adds 1u, right click removes 1uSplit 28-31: left click adds 1u, right click removes 1uSplit 29-32: left click adds 1u, right click removes 1uSplit 30-33: left click adds 1u, right click removes 1uCorner 28-29-31-32: left click adds 1u, right click removes 1uCorner 29-30-32-33: left click adds 1u, right click removes 1uSix line 28-33: left click adds 1u, right click removes 1u31 straight up: left click adds 1u, right click removes 1u32 straight up: left click adds 1u, right click removes 1u33 straight up: left click adds 1u, right click removes 1uSplit 31-32: left click adds 1u, right click removes 1uSplit 32-33: left click adds 1u, right click removes 1uStreet 31-33: left click adds 1u, right click removes 1uSplit 31-34: left click adds 1u, right click removes 1uSplit 32-35: left click adds 1u, right click removes 1uSplit 33-36: left click adds 1u, right click removes 1uCorner 31-32-34-35: left click adds 1u, right click removes 1uCorner 32-33-35-36: left click adds 1u, right click removes 1uSix line 31-36: left click adds 1u, right click removes 1u34 straight up: left click adds 1u, right click removes 1u35 straight up: left click adds 1u, right click removes 1u36 straight up: left click adds 1u, right click removes 1uSplit 34-35: left click adds 1u, right click removes 1uSplit 35-36: left click adds 1u, right click removes 1uStreet 34-36: left click adds 1u, right click removes 1uDozen 1-12: left click adds 1u, right click removes 1uDozen 13-24: left click adds 1u, right click removes 1uDozen 25-36: left click adds 1u, right click removes 1uColumn 1: left click adds 1u, right click removes 1uColumn 2: left click adds 1u, right click removes 1uColumn 3: left click adds 1u, right click removes 1u1 TO 18: left click adds 1u, right click removes 1uEVEN: left click adds 1u, right click removes 1uRED: left click adds 1u, right click removes 1uBLACK: left click adds 1u, right click removes 1uODD: left click adds 1u, right click removes 1u19 TO 36: left click adds 1u, right click removes 1u
Total stake0.00u0 active bet(s)
Positive outcome rate0.00%Any hit: 0.00%
EV / unit0.000uExpected net: 0.000u
House edge0.00%Push rate: 0.00%

Start by clicking a number or adding a common layout bet.