the math behind Roulette?

The short answer is no, the probability of red or black is not 50%. It is always slightly less, and that small difference is precisely how casinos guarantee their profits over time.

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1. The Probability of Red or Black

A roulette wheel has red pockets, black pockets, and green pockets (the zeros). The green pockets are neither red nor black. When the ball lands on green, even-money bets like Red and Black both lose.

The exact probabilities depend on the style of wheel:

European Wheel (Single Zero)

• Total pockets: $37$ (numbers $1$–$36$, plus a single green $0$)
• Red: $18$
• Black: $18$
• Green: $1$
• Probability of Red (or Black): $$P(\text{Red}) = \frac{18}{37} \approx 48.65\%$$

American Wheel (Double Zero)

• Total pockets: $38$ (numbers $1$–$36$, plus green $0$ and green $00$)
• Red: $18$
• Black: $18$
• Green: $2$
• Probability of Red (or Black): $$P(\text{Red}) = \frac{18}{38} \approx 47.37\%$$

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2. How the Casino Wins: Expected Value ($EV$) and the House Edge

The casino does not need to cheat or predict where the ball will land. They simply set the payout odds lower than the true odds.

When you bet on Red or Black, the casino pays out at $1:1$ (even money). If you bet $\$10$ and win, you get your $\$10$ back plus $\$10$ in profit.

Let’s calculate your expected value ($EV$) for a $\$1$ bet on an American wheel:

• Win: $+ \$1$ with probability $\frac{18}{38}$
• Lose: $- \$1$ with probability $\frac{20}{38}$ (the $18$ black pockets plus the $2$ green zeros)

$EV = \left( +1 \times \frac{18}{38} \right) + \left( -1 \times \frac{20}{38} \right) = \frac{18 - 20}{38} = -\frac{2}{38} \approx -\$0.0526$

On average, for every $\$1$ wagered, the player loses about $5.26$ cents. That difference is the House Edge:

(Notice that every extra green pocket nearly doubles the house's profit margin.)

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3. Does This Apply to Other Bets?

Yes. The house edge remains virtually identical across nearly every bet on the layout:

• Single Number Bet (Straight-up):
  • True probability of hitting your number (American): $\frac{1}{38}$
  • Fair payout: $37:1$
  • Actual casino payout: $35:1$
  • Expected return: $\left(35 \times \frac{1}{38}\right) + \left(-1 \times \frac{37}{38}\right) = -\frac{2}{38} = -5.26\%$

Whether you bet on a single number, split two numbers, choose a dozen, or bet Red/Black, the math is engineered so the casino retains the exact same mathematical edge.

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4. How the Casino Wins in Practice: The Law of Large Numbers

An individual player might walk up, bet $\$100$ on Red, win, and walk away ahead. In the short term, variance dominates.

However, a casino handles hundreds of thousands of spins across dozens of tables every month. According to the Law of Large Numbers, as the number of trials ($n$) grows:

$$\lim_{n \to \infty} \frac{\text{Actual Casino Revenue}}{\text{Total Money Wagered}} = \text{House Edge}$$

With enough spins, randomness smooths out, and the mathematical expectation functions essentially like a fixed tax on every dollar placed on the table.