Modeling the roulette ball as a probability experiment
A roulette ball turning in a plate can be modeled as a random experiment: after the ball loses speed, it drops into exactly one pocket. The standard introductory statistics assumption is that each pocket is equally likely.
Assumption (European roulette)
The wheel has 37 pockets labeled \(0,1,2,\dots,36\). There are 18 red pockets, 18 black pockets, and a single green pocket labeled \(0\). Each spin selects one pocket uniformly at random.
The sample space size is \(37\). For any event \(E\) defined by a set of favorable pockets, the basic probability rule is:
\[ P(E) = \frac{\text{number of favorable pockets}}{37}. \]
Common probabilities for roulette outcomes
Under the equal-likelihood model, probabilities are determined by counting.
| Event \(E\) | Favorable outcomes | Probability | Decimal |
|---|---|---|---|
| Ball lands on a specific number (e.g., 17) | 1 pocket | \(P=\dfrac{1}{37}\) | \(\approx 0.02703\) |
| Ball lands on red | 18 pockets | \(P=\dfrac{18}{37}\) | \(\approx 0.48649\) |
| Ball lands on black | 18 pockets | \(P=\dfrac{18}{37}\) | \(\approx 0.48649\) |
| Ball lands on 0 | 1 pocket | \(P=\dfrac{1}{37}\) | \(\approx 0.02703\) |
| Ball lands on an even number (2–36) | 18 pockets | \(P=\dfrac{18}{37}\) | \(\approx 0.48649\) |
| Ball lands on an odd number (1–35) | 18 pockets | \(P=\dfrac{18}{37}\) | \(\approx 0.48649\) |
Events such as “even” exclude \(0\) under standard casino rules; the single \(0\) pocket is the key reason probabilities are slightly below \(1/2\) for red/black and even/odd.
Expected value of typical roulette bets
Expected value measures the long-run average net gain per unit stake. For a 1-unit bet with win probability \(p\), net gain \(g\) on a win, and net loss \(-1\) on a loss, the expected value is:
\[ \mathbb{E}[\text{net}] = p\cdot g + (1-p)\cdot(-1). \]
Example 1: Straight-up bet on one number
A straight-up bet wins only if the roulette ball lands on the chosen number. In European roulette, \(p=1/37\). The standard payout is 35-to-1, meaning the net gain on a win is \(g=35\) units (stake returned separately).
\[ \mathbb{E}[\text{net}] = \frac{1}{37}\cdot 35 + \left(1-\frac{1}{37}\right)\cdot(-1) = \frac{35}{37} - \frac{36}{37} = -\frac{1}{37}. \]
Example 2: Even-money bet on red
A red bet wins if the ball lands on any red pocket. Here \(p=18/37\) and the net gain on a win is \(g=1\) unit.
\[ \mathbb{E}[\text{net}] = \frac{18}{37}\cdot 1 + \left(1-\frac{18}{37}\right)\cdot(-1) = \frac{18}{37} - \frac{19}{37} = -\frac{1}{37}. \]
House edge (European roulette)
For standard bets (straight-up, red/black, even/odd, dozens, columns), the expected value per 1 unit staked is \(-\dfrac{1}{37}\), which corresponds to a house edge of \[ \frac{1}{37}\times 100\% \approx 2.70\%. \]
Summary table of probability and expected value
| Bet type | Win probability \(p\) | Net gain on win \(g\) | \(\mathbb{E}[\text{net}]\) per 1 unit |
|---|---|---|---|
| Straight-up (one number) | \(\dfrac{1}{37}\) | \(35\) | \(-\dfrac{1}{37}\) |
| Red / Black | \(\dfrac{18}{37}\) | \(1\) | \(-\dfrac{1}{37}\) |
| Even / Odd (excluding 0) | \(\dfrac{18}{37}\) | \(1\) | \(-\dfrac{1}{37}\) |
| Dozen (1–12, 13–24, 25–36) | \(\dfrac{12}{37}\) | \(2\) | \(-\dfrac{1}{37}\) |
Visualization: pockets and equally likely outcomes
Final conclusions
Under the equal-likelihood model for a roulette ball turning in a plate on a European wheel, each pocket has probability \(1/37\), events such as red or even have probability \(18/37\), and the expected value of standard casino bets is \(-1/37\) of the stake per spin.