The analysis of a roulette system

The next scene on the CD-ROM shows the would-be systems player insisting that the computer simulation has confirmed the validity of the system. He'd tried it out several times and been able to survive for a free, one week holiday almost every time he'd run the program.

The money lender told him that, to be able to borrow money on such a system, he'd have to show conclusively that the system was infallible. When asked how to prove this infallibility, the money lender offers to create a computer program that will analyse the results of five thousand people going to the casino to play this system to try to win free holidays.

The reader of the CD-ROM is then presented with a program that simulates five thousand players going to the casino and using the system to win ten guineas a day for an indefinite period. In other words, each of the five thousand players stay playing until the day comes when the first seven spins of the roulette wheel corresponds to the random sequence they'd thought up that day on the way to the casino.

In this simulation, as each player makes daily bets, the results are recorded in a variety of ways. Not only is a record kept of how many days they survive before losing their money, but, the total number of bets and the amount accumulated. These vlues are displayed in fields on the screen. Further fields show the combined totals of all players' bets and amounts staked.

The money lender has run this program before the would be borrower returns to his office (Figure 2.1 shows his results).The dialogue goes like this:

Did you get that computer program finished?

Yes, and I left it running for several hours and have got the results of five thousand players playing your system.

What does it show you?

Here take a look.

There, what did I tell you? My system does work, 4647 out of the 5000 lasted more than a week.

Yes, but it also means that 353 lost their money in the first week. That's about seven percent of them: one in every fourteen players.

Then it isn't a perfect system?

No. In the long run, no system can succeed because the casino has an edge: it has the advantage of the zero in it's favour.

But how does this small advantage stop systems from winning?

It's due to the law of averages.

How does that apply?

With the zero in it's favour, the casino averages winning one thirty-seventh of all spins. If this is averaged out over a large number of spins, the casino wins one thirty-seventh of the total value of all bets made.

I still don't see how this advantage stops systems from working? Aren't systems supposed to overcome the house's advantage?

Take a look at the computer simulation results and tell me the average number of days holiday the players won.

One hundred and three.

Yes, but if they had each used their 1270 guineas to pay for their holidays at ten guineas a day they would each have had 127 days holiday.

Oh! I see. They have averaged out having less holidays by using my system?

That's right, they have each spent 1270 guineas and averaged only 103 days of holiday which works out that the average cost of each day's holiday using your system is over twelve guineas a day.

I see now why you have included the average cost of each day's holiday in your simulation.

Yes, it comes out on this simulation at 12.3301 guineas for every day of holiday won by gambling: making your roulette system an expensive way to pay for a holiday.

How is it then, that my system makes the holidays more expensive?

Take a look at the total number of spins that were bet on by the 5000 players.

Your results shows they bet on a total of 1,065,406 spins.

Yes over a million spins.

Now look at the total amount of money that they bet on those million spins.

Your results show 39,840,050 guineas that's almost exactly forty million guineas.

Now if you divide the total number of spins by the total amount bet.

You get 37.3942 guineas.

Yes. Because of the doubling up they have had to do, their average bet on each spin is about thirty seven and a half guineas. If we apply the casino's average take to this figure, that is one thirty-seventh, we should expect the casino to win an average of one thirty-seventh of this amount for every spin.

Let me work that out: 37.3940 divided by 37 gives 1.010655 - about one guinea a spin.

Yes the casino gains an average of just over a guinea per spin and, as you can see from the results, the players average just over 2 spins per day so the casino gains just over two guineas a day from each player.

So this is why the holidays are costing over twelve guineas a day with my system?

That's right. You cannot win with any system because the casino always wins approximately one thirty-seventh of every bet you make.

But using a system is better than no system at all.

No using a system like yours makes things worse.

Why?

Because the doubling up used by your system increases the total amount bet, so, the casino will win one thirty-seventh of a larger amount.

I don't understand that.

If you look at the simulation results again and divide the 40 million guineas bet by 5,000 you can work out that each player placed bets averaging a total of 8,000 guineas. The casino takes one thirty-seventh of this, which amounts to over 200 guineas per player.

So what is the best way to play roulette? What is the optimum system?

The way that minimizes the total amount bet. This means that the optimal way to play is to make one single bet of all your money, rather than spreading it out over a number of bets.

You mean there is no better system than going into a casino and placing all of your money on a single red?

Exactly. That is the optimum roulette system and there is a reward of ten thousand dollars for any system that can beat it.

But it cannot win?

Of course it cannot win. No system can win. Betting all your money on the red in a single bet simply loses less money over the long term.

I don't believe this.

I'll prove to you how much better it is to bet all of your money on the red rather than using the money to play your system.

How can you prove it?

I'll write another computer program to show what happens to 5,000 players when they play to win holidays using this optimum single bet system.

You mean you are going to run a simulation of 5,000 players, each going to Monte Carlo with 1270 guineas and playing to win or lose all with a single bet on red?

That's right, each one will either win enough for 254 days holiday, or, lose everything and have to go home without having had any holiday at all.

The next scenario sees the would be borrower returning, after this new computer simulation has been run:

Did you create that computer program to show what happens when players put all their money on the red?

Yes I ran it for 5,000 players again. Have a look at the results.

There are more losers than winners I notice.

Yes, but look how many holidays they averaged.

124 days each.

Yes that compares with the 103 days of holiday the players averaged using your system.

How did the single bet system produce better results?

Well look at the total amount that was bet by the single bet players.

Just over six million.

That's right. Now, do you remember how much the 5000 players bet using your system?

Almost forty million wasn't it?

Yes, using your system the casino gained one thirty-seventh of almost forty million, but, when players have just the one big bet on the red, the casino gains one thirty-seventh of only six million three hundred and fifty thousand.

Very clever. So you seem to have proved your point, a single bet on the red is better than my system.

I'm afraid so. Whatever system you come up with could never be better than the single bet system.

Of course, in chapter 1, you had probably realised that it wasn't the man playing roulette for a free holiday that was describing the optimum system for playing roulette. It was the other person, who, in the last line of the dialogue. spoke of preferring to make a single bet of all the money on red. No system of roulette, however complex can better this strategy. It doesn't win, but, it loses less than any other.