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The Roulette Odds Calculator computes exact probabilities, payouts, expected values, and house edge for all standard bet types on a European roulette wheel (37 pockets: 0-36). Understanding these mathematics is essential for informed decision-making and appreciating why the house always has a long-term advantage in roulette.
European roulette features a single zero pocket, giving a universal house edge of 2.70%. This is significantly better for players than American roulette, which adds a double zero (00) pocket, increasing the house edge to 5.26%. Every bet type in European roulette has the same house edge, making the game mathematically fair in relative terms across all wager types.
For European roulette with 37 pockets, the probability of winning a bet covering $$k$$ numbers is:
$$P(\text{win}) = \frac{k}{37}$$
The payout ratio for a bet covering $$k$$ numbers is:
$$\text{Payout} = \frac{36}{k} - 1$$
This means a winning bet returns the original wager plus the payout. The expected value (EV) per unit bet is:
$$EV = \frac{k}{37} \times \left(\frac{36}{k} - 1\right) \times B - \frac{37 - k}{37} \times B$$
Simplifying:
$$EV = B \times \left(\frac{36}{37} - 1\right) = -\frac{B}{37}$$
This means every bet has the same expected loss of $$B/37 \approx 2.70\%$$ of the wager, regardless of bet type. The house edge is:
$$\text{House Edge} = \frac{1}{37} \times 100\% \approx 2.703\%$$
The Win Probability shows your chance of winning on any single spin. The Payout Ratio shows how much you win per dollar bet (not including your original bet return). The Payout if Win includes your original bet. The Expected Value is always negative, representing the average loss per bet over the long run. The House Edge (2.70% for European) represents the casino's mathematical advantage. The Return to Player (97.30%) shows what percentage of total wagers are returned as winnings over time.
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A $10 straight-up bet on a single number wins with 2.70% probability, paying 35:1 ($360 total return). The expected loss is about $0.27 per spin.
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A $50 red/black bet wins 48.65% of the time (not 50% due to the green zero). The expected loss is $1.35 per spin, which is the same 2.70% house edge applied to a larger bet.
The green zero (0) pocket is neither red nor black. With 18 red, 18 black, and 1 green pocket out of 37 total, the probability of winning a red or black bet is $$18/37 \approx 48.65\%$$, not 50%. This 1.35% gap from 50% is what creates the house edge.
Mathematically, no. Every bet type in European roulette has the same house edge of $$1/37 \approx 2.70\%$$. Straight-up bets offer higher payouts but lower probability, while even-money bets offer lower payouts but higher probability. The expected loss per dollar wagered is identical.
European roulette has 37 pockets (0-36) with a 2.70% house edge. American roulette has 38 pockets (0, 00, 1-36) with a 5.26% house edge. The extra pocket nearly doubles the casino's advantage. Always choose European if available.
No betting strategy (Martingale, Fibonacci, D'Alembert, etc.) can overcome the mathematical house edge in the long run. These systems may change the distribution of wins and losses (more frequent small wins vs. rare large losses) but cannot change the expected value of $$-B/37$$ per bet.
Expected value is the average outcome per bet over many repetitions. It is negative because the payout ratios are based on 36 (as if there were 36 pockets) while the actual wheel has 37 pockets. This discrepancy of one pocket is the source of the house edge: $$EV = -B/37$$.
Over 100 spins betting $$B$$ each time, your expected total loss is $$100 \times B/37 \approx 2.70 \times B$$. For $10 bets, expect to lose about $27 over 100 spins. Actual results will vary significantly due to randomness, but this is the statistical expectation.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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