Playing the roulette wheel of (mis)fortune

It is well known that playing casino games can lead to ruin (at least financial ruin) if the gambler is not careful (e.g. putting all or a substantial portion of his/her wealth on the line with the goal of beating the casino). There are two ways to prove that such notion is foolish and reckless for anyone that needs to be convinced. One is to through empirical data and the other is through mathematics. Both ways show that the fate of the gambler is indeed grim.

The empirical data can be obtained in two ways. One is bringing real money into a real casino to gamble. The other is computer simulation. The first way is at best an expensive way to collect data and at worse putting your financial well being at risk. Game results can be generated using computer generated random numbers based on the presumed odds of the game in question. Since no real money is involved, it is easy to generate thousands or tens of thousands or more bets (or millions of bets if you wish). It is a cheap way to generate results that mimic making a large number of bets. All one needs is a software program that has a random number generator.

Let’s simulate results from a roulette wheel (American roulette). In the American version, the wheel has 38 slots – 0, 00 and the numbers 1 through 36.The two slots 0 and 00 are green and the numbers 1 to 36 are half red and half black. The following shows the roulette wheel and the table layout.

Players may choose to place bets on either a single number or a range of numbers, the color red or black, or whether the number is odd or even, or if the numbers are high (19–36) or low (1–18).

Let’s focus on the bet on the color red. Note that there are 18 red slots out of 38. The payout on this bet is 1 to 1 (if the ball lands on any of the 20 non-red slots, the player loses one unit and if the ball lands on a red slot, the player wins one unit). The odds for winning (for the player) are 18 to 20. Turning it around, the odds for losing (for the player) are 20 to 18. The odds for winning for the player are not 1 to 1 but the payout is 1 to 1. This means that the house has an edge.

Essentially, the green slots 0 and 00 give the edge to the house. These two slots give the player extra two possibilities to win or the house two extra possibilities to win. For example, for 38 bets of $1 each on the color red, the player is expected to win 18 times (gain $18) and lose 20 times (lose $20) with an overall loss of $2. In fact, no matter what the bet is (bet on color, bet on even-odd, bet on a range of numbers or a single number), the payout rule is designed so that the player is, on average, expected to lose 2 bets per 38 bets. Since 2/38 = 0.0526 (5.26%), this gives the house the edge of 5.26%. For a more detailed discussion of the roulette game, see this post from a companion blog.

If anyone who still does not believe the math, maybe a computer simulation can help. The simulation is done in Excel, which has functions for generating random numbers that can be very handy. For the purpose at hand, we use the function =RANDBETWEEN(a, b), which gives a random number between a and b.

In each turn of the roulette wheel, the ball can land in any one of the 38 slots. So we use =RANDBETWEEN(-1, 36). If the result is -1 or 0, we take that as the result of landing in a green slot (-1 for 00 and 0 for 0). If the result is in 1 to 36, we just take it as the result of landing in the slot with that number.

We generate 10,000 random numbers using =RANDBETWEEN(-1, 36). To generate 10,000 values, put the formula in cell A1 and then copy the formula down to cell A10000. Next convert these 10,000 numbers from formulas into values (by copying these 10,000 cells and then pasting onto the same cells as values). The following table shows the results on the first 10 plays.

Table 1 – First 10 Simulated Roulette Games (bets on Red with $1 per bet)

Random Number Color Player’s Winning
32 red $1
9 red $1
25 red $1
8 black -$1
1 red $1
36 red $1
13 black -$1
3 red $1
14 red $1
23 red $1

The roulette player is doing well in the first 10 games (8 wins and 2 losses) with $6 in winnings. Here’s the next 10 simulated plays.

Table 2 – Next 10 Simulated Roulette Games (bets on Red with $1 per bet)

Random Number Color Player’s Winning
-1 green -$1
18 red $1
26 black -$1
30 red $1
29 black -$1
3 red $1
33 black -$1
1 red $1
21 red $1
29 black -$1

In the second 10 games, the roulette player is just making even with 5 red numbers and 5 non-red numbers. In the first 20 simulated games, the ball lands on 13 red slots and 7 non-red slots for the total winning of $6. So in the first 20 simulated games, the player has an edge, winning $0.30 per bet (6/20 = 0.30). As the simulation progresses, the results for the player keeps getting progressively worse.

Table 3 – Summary of the 10,00 Simulated Roulette Games

First n Games Number of Red Balls Winning Amounts Average Winning per $1 Bet
20 13 $6 $0.30
100 57 $14 $0.14
1,000 484 -$32 -$0.032
5,000 2,371 -$258 -$0.0516
10,000 4,738 -524 -$0.0524

In the first 100 games, the player comes out ahead winning $14 (or 14 cents per $1 bet). If he or she stops right there, the player will bring home real money. If the player thinks that he or she will continue to have 14% edge in his/her favor, he or she is mistaken. At the end of the simulation, the average winning is negative $0.0524, very close to the 5.26% house edge discussed earlier.

One thing that should be pointed out is that in actual playing, the player will have to stop long before 10,000 games for the reason of running out of money. For the players that carry only cash into the casino, he or she will run out of cash. For example, assuming that the player starts with $100, the player will run out of cash at the 1372th simulated game in the simulation discussed above. That is a long series of games (being over 1,000 games long). The point is that the rules are stacked against the player. If someone plays long enough, the result is ruin.

Of course, simulations are random (just as actual games of chance are random). Another simulation will produce different results. But the overall pattern will be the same. We generate a few more simulations of 10,000 games each. The following table shows the results.

Table 4 – A Few More Simulations, 10,000 Games each

Simulation # Number of Red Balls Winning Amounts Average Winning per $1 Bet
1 4,691 -$618 -$0.0618
2 4,727 -$546 -$0.0546
3 4,767 -$466 -$0.0466
4 4,718 -$564 -$0.0564
5 4,640 -$720 -$0.0720
6 4,780 -$440 -$0.0440
7 4,713 -$574 -$0.0574
8 4,807 -$386 -$0.0386
9 4,788 -$424 -$0.0424
10 4,739 -$522 -$0.0522

There is indeed a great deal of fluctuation in the results in table 4. Some have more red balls than the others. The average winnings range from negative 3 cents to negative 7 cents per $1 bet.

Another thing to point out is that the number of red balls in Table 4 gives the empirical winning odds for the red ball bet. Recall that the theoretical winning odds for the bet on red ball are 18 in 20 (18 possibilities in winning for the player vs. 20 possibilities in winning for the house). Since 20/18 = 1.11, the winning odds for the player are 1 in 1.11. For each of the simulations in Table 4, the odds are roughly 1 in 1.11. For example, the odds for the first simulation in Table 4 are 4691 in 5309. This translates to 1 in 1.13 (5309/4691 = 1.13).

When performing a random experiment (e.g. making bets at the roulette table), the individual observations are not predictable. However, the long run average of many independent observations is predictable and stable. This is called the law of large numbers. For example, there is no way one can predict whether a player will lose on any given bet at the roulette table. In any one bet, the player may win or lose. In the first 10 simulated games of roulette described in Table 1, the player does quite well. But in the long run, the player will lose (and the casino wins).

According to the law of large numbers, the average results of a random experiment approach the theoretical average. The theoretical winning for the player is negative 5.26% per unit amount. The simulations show that there is a great deal of fluctuation in the individual simulated games. Table 4 shows that the average winnings in the long run do hover around 5.26%.

Playing games of chance for money can be a great entertainment, but only if playing in such a way that the loss is limited and that the amount of potential loss is money that you feel you can afford to lose. That amount is essentially the cost of the entertainment.

Similar simulations are discussed in this post in the context of the carnival game Chuck-a-Luck.