Roulette table as a probability model
A roulette table represents a finite set of numbered outcomes. Under the standard assumption of a fair wheel, each slot is equally likely. Probability calculations reduce to counting outcomes:
If the roulette wheel has \(N\) equally likely slots and an event \(A\) contains \(k\) favorable slots, then \[ P(A) = \frac{k}{N}. \]
To make the computations concrete, assume an American roulette table with \(N=38\) slots: \(\{0,00,1,2,\dots,36\}\). (For a European roulette table, \(N=37\) with only \(\{0,1,\dots,36\}\); the same steps apply.)
Visualization: simplified roulette table layout
Step-by-step: probability from a roulette table
- Step 1: Identify the wheel type and total slots \(N\) (American \(N=38\), European \(N=37\)).
- Step 2: Identify which slots win for the bet (favorable outcomes).
- Step 3: Count favorable outcomes \(k\) and compute \(P(\text{win})=\dfrac{k}{N}\).
- Step 4: Use the payout rule to compute expected value for a \$1 bet.
Expected value and house edge (using the roulette table)
For a \$1 bet, define the net profit as \(+w\) dollars on a win and \(-1\) dollar on a loss (the \$1 stake is lost). If \(P(\text{win})=p\), then:
\[ E(\text{profit}) = p\cdot w + (1-p)\cdot(-1). \]
Under standard roulette payouts on an American roulette table, many common bets yield the same expected profit: \[ E(\text{profit}) = -\frac{2}{38} = -\frac{1}{19} \approx -0.05263 \text{ per \$1 bet}. \] This corresponds to a house edge of about \(5.263\%\).
Worked examples
Example 1: Straight-up bet on a single number
A straight-up bet wins if exactly one chosen number occurs. On an American roulette table, \(k=1\) favorable outcome and \(N=38\).
\[ p = P(\text{win}) = \frac{1}{38}. \]
Standard payout is \(35\) to \(1\), so net profit \(w=35\). Then:
\[ E(\text{profit})=\frac{1}{38}\cdot 35 + \left(1-\frac{1}{38}\right)\cdot(-1) = \frac{35}{38}-\frac{37}{38}=-\frac{2}{38}. \]
Example 2: Even-money bet (red/black or even/odd)
On an American roulette table, there are \(18\) outcomes in the chosen group (for example, “red”), \(18\) in the opposite group, and \(2\) green outcomes (\(0\) and \(00\)) that lose for both groups. Thus \(k=18\), \(N=38\), so \(p=\dfrac{18}{38}\).
Even-money payout is \(1\) to \(1\), so \(w=1\). Then:
\[ E(\text{profit})=\frac{18}{38}\cdot 1 + \left(1-\frac{18}{38}\right)\cdot(-1) =\frac{18}{38}-\frac{20}{38}=-\frac{2}{38}. \]
Common roulette bets summarized (American roulette table)
The table below uses \(N=38\) total outcomes. “Favorable outcomes” is the number of slots covered by the bet, and “net win \(w\)” is the profit on a \$1 win (not including returning the stake).
| Bet type | Favorable outcomes \(k\) | \(P(\text{win})=\dfrac{k}{38}\) | Net win \(w\) | Expected profit for \$1 |
|---|---|---|---|---|
| Straight-up (1 number) | 1 | \(\dfrac{1}{38}\) | 35 | \(-\dfrac{2}{38}\) |
| Split (2 numbers) | 2 | \(\dfrac{2}{38}\) | 17 | \(-\dfrac{2}{38}\) |
| Street (3 numbers) | 3 | \(\dfrac{3}{38}\) | 11 | \(-\dfrac{2}{38}\) |
| Corner (4 numbers) | 4 | \(\dfrac{4}{38}\) | 8 | \(-\dfrac{2}{38}\) |
| Six line (6 numbers) | 6 | \(\dfrac{6}{38}\) | 5 | \(-\dfrac{2}{38}\) |
| Dozen (12 numbers) | 12 | \(\dfrac{12}{38}\) | 2 | \(-\dfrac{2}{38}\) |
| Column (12 numbers) | 12 | \(\dfrac{12}{38}\) | 2 | \(-\dfrac{2}{38}\) |
| Red/Black (18 numbers) | 18 | \(\dfrac{18}{38}\) | 1 | \(-\dfrac{2}{38}\) |
| Even/Odd (18 numbers) | 18 | \(\dfrac{18}{38}\) | 1 | \(-\dfrac{2}{38}\) |
| High/Low (18 numbers) | 18 | \(\dfrac{18}{38}\) | 1 | \(-\dfrac{2}{38}\) |
Adapting the same method to a European roulette table
For a European roulette table, replace \(38\) by \(37\) and remove the \(00\) slot. For example, a red/black bet has \(k=18\) favorable outcomes and \(N=37\), so \(P(\text{win})=\dfrac{18}{37}\). With standard even-money payout \(w=1\), \[ E(\text{profit})=\frac{18}{37}\cdot 1 + \left(1-\frac{18}{37}\right)\cdot(-1)= -\frac{1}{37}\approx -0.02703, \] giving a house edge of about \(2.703\%\).
Quick checklist for roulette table probability
- Count outcomes: determine \(N\) from the roulette table type.
- Count favorable slots: determine \(k\) from how the bet covers the table.
- Compute probability: \(P=\dfrac{k}{N}\).
- Compute expected profit: \(E=p\cdot w+(1-p)\cdot(-1)\).