## Roulette

At the Crown Casino in Melbourne, Australia, some roulette wheels have 18 slots colored red, 18 slots colored black, and 1 slot (numbered 0) colored green.  The red and black slots are also numbered from 1 to 36. (Note that some of the roulette wheels also have a double zero, also colored green, which nearly doubles the house percentage.)

You can play various ‘games’ in roulette.  Consider:

A. Betting on Red

This game involves just one bet.  You bet \$1 on red.  If the ball lands on red you win \$1, otherwise you lose.

B.  Betting on a Number

This game involves just one bet.  You bet \$1 on a particular number, say 17; if the ball lands on that number you win \$35, otherwise you lose.

C.  Martingale System

In this game you start by betting \$1 on red.  If you lose, you double your previous bet; if you win, you bet \$1 again.  You continue to play until you have won \$10, or the bet exceeds \$100.

D.  Labouchere System

In this game you start with the list of numbers (1,2,3,4).  You bet the sum of the first and last numbers on red (initially \$5).  If you win you delete the first and last numbers on red (so if you win your first bet it becomes (2,3)), otherwise you add the sum to the end of your list (so if you lose your first bet it becomes (1,2,3,4,5)).  You repeat this process until your list is empty, or the bet exceeds \$100.  If only one number is left on the list, you bet that number.

1.  Simulation

For each game, we write a function (with no inputs) in R that plays the game once and returns a vector of length two consisting of the amount won/lost and how many bets were made.  (Edit: Labouchere function corrected)

R Code: Roulette functions

Using these functions, we now simulate 100,000 repetitions of each game and estimate 1. expected winnings per game; 2. proportion of games you win; 3. expected playing time per game.

R Code: Roulette Simulation

A. Betting on Red

1. $E(winnings) \approx -.025$

2. $\Pr(win) \approx .4876$

3. $E(bets) \approx 1$

B. Betting on a Number

1. $E(winnings) \approx -.039$

2. $\Pr(win) \approx .02695$

3. $E(bets) \approx 1$

C. Martingale System

1. $E(winnings) \approx -2.06$

2. $\Pr(win) \approx .9096$

3. $E(bets) \approx 19.5256$

D. Labouchere System

1. $E(winnings) \approx -2.67$

2. $\Pr(win) \approx .953$

3. $E(bets) \approx 8.96$

2.  Verification

For games A and B, we now calculate the theoretical values for 1. expected winnings per game; 2. proportion of games you win.

A.  Betting on Red

1. $E(winnings) = -.027$

percentage error of estimate $= 7.94\%$

2. $\Pr(win) = .486$

percentage error of estimate $= .22\%$

B.  Betting on a Number

1. $E(winnings) = -.027$

percentage error of estimate $= 10.26\%$

2. $\Pr(win) = .027$

percentage error of estimate $= 85.2\%$

Now we calculate explicitly the maximum amount you can lose with each strategy.  Let $L$ denote the amount lost playing each system.

A.  Betting on Red

Trivially, $\max(L) = 1$

B.  Betting on a Number

Trivially, $\max(L) = 1$

C.  Martingale System

Since winning any bet under this system brings the net winnings up to \$1 and resets the bet to \$1, the maximum loss situation is the one in which the player loses every bet he plays until the bet exceeds \$100.  Thus, $\max(L) = 127$

D.  Labouchere System

Similar to the Martingale System, winning any bet in the Labouchere System decreases the maximum possible loss.  Therefore, the maximum loss situation is the one in which the player loses every bet he plays until the bet exceeds \$100.  This is the sum of the numbers from 5 to 100: $\max(L) = 5040$

3.  Variation

We now wish to repeat the simulation experiment from Part 1 five times to obtain sample minimum and maximum values for expected winnings, proportion of games won, and expected playing time.  To do this, we loop the experiment 5 times, storing our values in vectors, and later take minimum and maximum values from them.  ( A side note: As I write this, I have started this simulation,  and realize this is definitely not the fastest/most efficient way to go about things.  But enough time has passed that (for example) installing a new package and modifying the code to utilize parallel for loops, and THEN running a simulation would take just as long.)

R Code for Variation Simulation We can see from these results that all simulations had negative expecations for all simulations.  It is interesting to note that both the Labouchere and the Martingale system win a high percentage of the time, yet end up with negative expectations.  The Martingale system has most expected bets per game.

Now we modify this code again, so that in addition to estimating the expected winnings, expected proportion of wins, and expected playing time, it also estimates the standard deviation of each of these values.  (Note: since we only run this experiment once (with 100,000 trials), sequential run-time is sufficient.)

R Code for Variation (Mean and Std. Dev.) Reviewing these results we see that the expected amount won is most variable in the Labouchere System (by FAR).  The expected playing time is also most variable in the Labouchere System (although the Martingale system had the highest mean playing time).

In summary, none of these systems give the player an advantage over the casino. However, we do notice some interesting things about the Martingale system and the Labouchere system: while the expected winnings are negative, the player wins a high percentage of the time $(> 90\%)$.  How do we explain this phenomenon?  In the games the player loses money, he loses such a massive amount that it wipes out all of his previous winnings (and then some).

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