Let's Play Roulette - Just for Fun

Let's Play Roulette - Just for Fun

June 14, 2012

The other day it occurred to me I could double my money at Roulette if only I was willing to play a large enough numbers of times, even with possible losses building up before the ultimate win.  So, I decided to try this by simulation.

Here’s the game.  You have a table with 36 number slots, half red and half black with two additional green slots, zero and double zero.  This gives a total of 38 possible outcomes for American roulette.  The table is circular, something like a bowl. A steel marble is sent a spinning in one direction and the table is sent spinning in the opposite direction.  When everything slows down, the marble settles into one of the slots.  That number pertaining to this slot is the winner.   You can bet on any number, or perhaps bet a red or black number will come up. In our game we will always play red. The payoff for any number is 36:1, and the payoff for red is your bet, 1:1.   We always bet the same amount of $1 on red.    More detail on roulette can be found on Wikipedia at http://en.wikipedia.org/wiki/Roulette

Facts: the probability of winning on a fair table is 18/38 = 0.473684211, just slightly less than 0.50.  Those green slots make up for the difference.   Not much less you think, right? 

Goal #1:  Double my money.  That is, I play until I have $2 – doubling what I started with.   I played roulette on my simulator 100 separate rounds for this goal.  The result is that I needed to play an average of 18 spins (about one hour) to win the dollar, when it happened.  During this set of spins, I had a maximum loss of $23.  But, and this is a huge huge, about 9% of the time the game went more than 1.5 million spins (where I stopped playing) without doubling.  In these cases my average loss was about $80,000.  

Goal #2: Quadruple my money.  That is, I play until I have $4 – quadrupling what I started with. I played roulette on my simulator 100 separate rounds for this goal.  The result is I needed to play an average 42 spins (about two hours) to quadruple my money – when it happened.  But about 25% of the time the game went more than 1.5 million spins (where I stopped playing) without quadrupling.  In these cases my average loss was (still) about $80,000.

In the vast majority of both these simulations, the winning happened right off basically in just a few spins, respectively.   Once you get too far negative, the probability of coming back is extremely small.

Goal #3: Multiply my initial money by ten. That is, I play until I have $10 – 10-tupling what I started with.  In this case, we 10-tupled our money only 33% of the time, the game terminated at 1.5 million spins about 67% of the time.

Note.  Altogether, in these three examples, we “spun” the steel marble 150,009,980 times.  In real life, assuming the croupier can make 20 spins/hour, and you can stand at the table 24/7, this would take a mere 856 years.  You would have lost about $7,864,206 for your trouble, probably a little sleep, and a few pounds, as well.

By removing those two green slots and making the game completely fair, the phenomena of game termination only rarely occurs at any n- tupling level, 2, 4, or 10.  But at the 100-tuple level it does occur again with measurable fraction of the time.  Maybe I should deliver a comprehensive table of data for all this.

The moral of this story is that you're unlikely to make money, much let get rich playing roulette.  If you do have to play, assign yourself a budget (stake) and a goal.  Quit when you have lost your stake!  You have just paid for the game's entertainment value - and maybe a little titillation.

BTW, there are applications of all this to physics, and there is some serious mathematics that does give expressions for the underlying probabilities based on semi-infinite random walks or Markov chain ideas.  Indeed, using Markov chains, though finite, these simulations are essentially validated.  Finally, simulations are extremely accurate and much more fun.  Did you know, nuclear reactor design is somewhat based on numerical simulations?!