
## 5. Roulette

### The Roulette Wheel

The (American) roulette wheel has 38 slots numbered 00, 0, and 1-36. As shown in the picture below,

• slots 0, 00 are green;
• slots 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 are red;
• slots 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 are black.

Except for 0 and 00, the slots on the wheel alternate between red and black. The strange order of the numbers on the wheel is intended so that high and low numbers, as well as odd and even numbers, tend to alternate.

According to Richard Epstein, roulette is the oldest casino game still in operation. It's invention has been variously attributed to Blaise Pascal, the Italian mathematician Don Pasquale, and several others. In any event, the roulette wheel was first introduced into Paris in 1765.

The roulette experiment is very simple. The wheel is spun and then a small ball is rolled in a groove, in the opposite direction as the motion of the wheel.. Eventually the ball falls into one of the slots. Naturally, we assume mathematically that the wheel is fair, so that the random variable $$X$$ that gives the slot number of the ball is uniformly distributed over the sample space $$S = \{00, 0, 1, \ldots, 36\}$$. Thus, $$\P(X = x) = \frac{1}{38}$$ for each $$x \in S$$.

### Bets

As with craps, roulette is a popular casino game because of the rich variety of bets that can be made. The picture above shows the roulette table and indicates some of the bets we will study. All bets turn out to have the same expected value (negative, of course).

#### Straight Bets

A straight bet is a bet on a single number, and pays $$35 : 1$$.

Let $$W$$ denote the winnings on a unit straight bet. Then

1. $$\P(W = -1) = \frac{37}{38}$$, $$\P(W = 35) = \frac{1}{38}$$
2. $$\E(W) = -\frac{1}{19} \approx -0.0526$$
3. $$\sd(W) \approx 5.7626$$

#### Three Number Bets

A 3-number bet (or row bet) is a bet on the three numbers in a vertical row on the roulette table. The bet pays $$11 : 1$$.

Let $$W$$ denote the winnings on a unit row bet. Then

1. $$\P(W = -1) = \frac{35}{38}$$, $$\P(W = 11) = \frac{3}{38}$$
2. $$\E(W) = -\frac{1}{19} \approx -0.0526$$
3. $$\sd(W) \approx 3.2359$$

#### Six Number Bets

A 6-number bet or 2-row bet is a bet on the 6 numbers in two adjacent rows of the roulette table. The bet pays $$5 : 1$$.

Let $$W$$ denote the winnings on a unit 6-number bet. Then

1. $$\P(W = -1) = \frac{16}{19}$$, $$\P(W = 5) = \frac{3}{19}$$
2. $$\E(W) = -\frac{1}{19} \approx -0.0526$$
3. $$\sd(W) \approx 2.1879$$

#### Eighteen Number Bets

An 18-number bet is a bet on 18 numbers. In particular, A color bet is a bet either on red or on black. A parity bet is a bet on the odd numbers from 1 to 36 or the even numbers from 1 to 36. The low bet is a bet on the numbers 1-18, and the high bet is the bet on the numbers from 19-36. An 18-number bet pays $$1 : 1$$.

Let $$W$$ denote the winnings on a unit 18-number bet. Then

1. $$\P(W = -1) = \frac{10}{19}$$, $$\P(W = 1) = \frac{9}{19}$$
2. $$\E(W) = -\frac{1}{19} \approx -0.0526$$
3. $$\sd(W) \approx 0.9986$$

In the roulette experiment, select the 18-number bet. Run the simulation 1000 times and watch the apparent convergence of the empirical density function and moments of $$W$$ to the true probability density function and moments. Suppose that you bet \$1 on each of the 1000 games. What would your net winnings be?

Although all bets in roulette have the same expected value, the standard deviations vary inversely with the number of numbers selected. What are the implications of this for the gambler?