How the Calculator Works

Roulette captures the imagination because of its elegant simplicity. A croupier launches a small ball around a spinning wheel that contains numbered pockets. Where the ball lands determines who wins. Behind this spectacle lies a set of well-defined probabilities that govern every possible wager. Understanding those probabilities allows players to make informed decisions about risk and reward. The calculator above lets you explore the mathematics of common roulette bets without needing any external references. By specifying the type of wheel and bet, you receive the exact probability of winning, the payout multiple, and the expected return for a single unit wager.

The essential difference between European and American roulette is the number of pockets. A European wheel contains numbers 1 through 36 and a single zero, for a total of 37 pockets. An American wheel adds a double zero, increasing the count to 38. That extra slot alters the house edge for every bet. The formula for the probability of a win is straightforward: divide the number of pockets covered by your bet by the total pockets on the wheel.

For example, a straight bet on a specific number covers just one pocket. On a European wheel the probability is therefore $\frac{1}{37}$, whereas on an American wheel it is $\frac{1}{38}$. More expansive bets, such as red/black, cover eighteen pockets, so their probabilities are $\frac{18}{37}$ or $\frac{18}{38}$ depending on wheel type.

The expected value (EV) for a bet expresses the average gain or loss per unit wager if the bet were repeated infinitely. It combines the chance of winning with the payout offered. Roulette payouts follow longstanding conventions: a winning straight bet pays 35:1, a split pays 17:1, a street pays 11:1, a corner pays 8:1, a six line pays 5:1, dozens and columns pay 2:1, and even-money bets like red/black pay 1:1. To compute EV we subtract the probability of losing from the product of win probability and payout:

$\mathrm{EV}=P\left(\mathrm{win}\right)\times \mathrm{Payout}-(1-P(\mathrm{win}\left)\right)$

A negative EV indicates an average loss to the player, which is the norm in casino games. By comparing EVs, we see how the house edge manifests in different wagers.

Table of Common Bets

The table below summarizes coverage, payout, and probability for standard bets. The final column lists the house edge for each wager, calculated as the negative of the EV for a one-unit bet. Notice that for both wheels the edge is constant across bets even though probabilities differ; this uniformity is a hallmark of roulette design.

Bet	Numbers Covered	Payout	Prob. Win (EU)	Prob. Win (US)	House Edge
Straight	1	35:1	1/37	1/38	≈2.7% / 5.26%
Split	2	17:1	2/37	2/38	≈2.7% / 5.26%
Street	3	11:1	3/37	3/38	≈2.7% / 5.26%
Corner	4	8:1	4/37	4/38	≈2.7% / 5.26%
Six Line	6	5:1	6/37	6/38	≈2.7% / 5.26%
Dozen/Column	12	2:1	12/37	12/38	≈2.7% / 5.26%
Red/Black Odd/Even High/Low	18	1:1	18/37	18/38	≈2.7% / 5.26%

Using Probability to Inform Strategy

Despite the uniform house edge, the volatility of different bets varies dramatically. Straight bets hit rarely but pay handsomely when they do; even-money bets succeed about half the time but only double your stake. Understanding this trade-off helps players align choices with their risk tolerance. Our calculator reports both the raw probability and the EV so you can quantify the difference. Suppose you wager one unit on a straight bet in European roulette. The probability of winning is about 2.70%, and the EV is −0.0270. That means that over many spins you would expect to lose 2.7% of each unit wagered. Switching to red/black retains the same EV but yields a 48.6% chance of winning, smoothing the ride at the expense of potential large payouts.

The American wheel carries a larger house edge because of the additional double-zero pocket. For a straight bet, the probability drops to 1/38, and the EV becomes roughly −0.0526. In other words, the casino expects to keep about 5.26% of the total amount wagered on average. While casual players may not feel the difference in the short term, over time the extra edge compounds. That is why serious roulette players seek out European wheels whenever possible.

Probabilities also interact with bankroll management. Consider a player starting with 100 units and betting one unit at a time on red in European roulette. After 100 spins, the expected loss is 100 × 0.027 = 2.7 units, leaving 97.3 units on average. Variance means the actual result may be higher or lower, but the EV provides an unbiased estimate.

Mathematical Derivations

The probability model for roulette is built on basic combinatorics. For each spin, the sample space contains N equiprobable outcomes where N is the number of pockets. A bet covering k numbers has winning probability $\frac{k}{N}$. The payout ratio r is the amount won per unit stake if the bet succeeds. A single play is a Bernoulli trial with success probability p and reward r. The expected return is therefore $p\times r-(1-p)$. Substituting $p=\frac{k}{N}$ yields $\frac{k}{N}\times r-(1-\frac{k}{N})$. Because payouts are set so that $r=\frac{N}{k}-1$, simplification confirms that the house edge remains constant across bets: $\frac{1}{N}$ for European and $\frac{2}{N}$ for American. The derivation illustrates how the game is balanced to ensure the same expected loss regardless of bet complexity.

Example Calculation

Imagine a player choosing a corner bet on a European wheel. A corner covers four adjacent numbers, so the probability of winning is 4/37 ≈ 10.81%. The payout is 8:1, meaning the player gains eight units plus the original stake when successful. Using the EV formula we obtain $\frac{4}{37}\times 8-(1-\frac{4}{37})$ ≈ −0.027. If the player places this bet repeatedly, they will lose about 2.7% of the wagered amount in the long run, identical to other European bets.

For an American wheel, a corner bet still covers four numbers but the denominator becomes 38. The EV changes to approximately −0.0526, highlighting the greater cost of playing on the double-zero wheel. Switching to a red/black bet reduces volatility but not the expected loss. Regardless of approach, roulette remains a negative expectation game.

Responsible Gaming

While mathematics can illuminate the structure of roulette, it cannot guarantee profits. The house edge ensures that casinos win over time. Players should treat roulette as entertainment and wager only what they can afford to lose. Establishing win and loss limits, taking breaks, and avoiding chasing losses are prudent habits. Understanding odds can make the experience more engaging and transparent, but it does not eliminate risk. This calculator aims to demystify the numbers so that you enter the game with open eyes.

Roulette’s enduring appeal lies in its blend of chance, tradition, and spectacle. With each spin, probabilities reset; past outcomes do not influence future ones. Nonetheless, exploring the mathematics can deepen appreciation for the design. Whether you favor risky straight bets or conservative even-money wagers, knowing the odds helps you gauge excitement against expectation. Use the tool above to experiment with different scenarios, compare wheel types, and understand how subtle changes in pocket count affect the bottom line.