# Gambling and the law of large numbers

## Main Question or Discussion Point

Does the law of large numbers really apply directly to betting systems? For example, in American roulette the house advantage or "edge" is 5.26%, and smart players know that, as a consequence of the law of large numbers, you will lose 5.26 cents of every dollar bet in the long run. This is supposed to apply no matter what ingenious betting system you use, as long as there is a maximum bet and you keep playing indefinitely.

The statement of the strong law of large numbers usually applies to a sum of independent random variables. But the winnings from each trial are not independent, even though the trial outcomes themselves are. This is because the bet sizes on different trials are not independent for most any "system". So how do you know that the law of large numbers applies to the total winnings after n trials?

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mathman
The law of large can be generalized to cover dependent random variables. The limit on the bet size insures that it can be applied.

The law of large can be generalized to cover dependent random variables. The limit on the bet size insures that it can be applied.

That is good news. Can you give a hint as to how the normal proof for independent variables has to be modified, or a book reference that covers it? Thanks.

Have you tried out a number of different systems like those described here?: http://www.lolroulette.com/roulette/systems/

None of the systems actually work when you factor in a large number of systems. The edge is 5.26% in American roulette and 2.70% in European roulette (which is a better game for winning by the way). Each spin is independent of the previous no matter what system you are using.

It turns out that an American roulette wheel has 38 slots. Players can wager either on red or black, which are both in equal numbers (18 of each, 36 total). The odds of winning on this would normally be 50/50 (like flipping a coin) but since this wheel has TWO green slots, it means that every so often, you will lose whether you are betting on red or black. These green slots are essentially the casino's profit margins

So the edge comes from 2/38 or 5.26%. Which is the profit the casino takes in over the long run. I hope this what you were looking for.

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mathman
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Nice! Thanks for finding that. All but the dumbest systems are gonna have the bet sizes getting less and less correlated as |i-j|--> infty. But just to show that a system could fail to satisfy this condition, consider "The Guv'nor" (ok, I just made that up): Bet the minimum on the first trial. If the first trial is a win, continue betting the minimum on every trial thereafter. If the first trial is a loss, bet the maximum on every subsequent trial. Continue until broke. :)

Have you tried out a number of different systems like those described here?: http://www.lolroulette.com/roulette/systems/

None of the systems actually work when you factor in a large number of systems. The edge is 5.26% in American roulette and 2.70% in European roulette (which is a better game for winning by the way). Each spin is independent of the previous no matter what system you are using.

It turns out that an American roulette wheel has 38 slots. Players can wager either on red or black, which are both in equal numbers (18 of each, 36 total). The odds of winning on this would normally be 50/50 (like flipping a coin) but since this wheel has TWO green slots, it means that every so often, you will lose whether you are betting on red or black. These green slots are essentially the casino's profit margins

So the edge comes from 2/38 or 5.26%. Which is the profit the casino takes in over the long run. I hope this what you were looking for.
Thanks for that link. It describes the exact systems I had in mind when I started this thread, so it saves me the trouble. I must confess that I have never even played roulette or any other casino game where these systems could be applied. The one time I was in a casino (for a league pool tourney), I lost \$20 at video poker slots and got bored with it quickly. My gambling experience is limited mostly to pool halls and bars.

I had "known" for a long time that these betting sytems are worthless except for their entertainment value. I was satisfied with the intuitive argument that since each individual bet has negative expected value, and one spin cannot influence another, then there's nothing you can do to win in the long run. That argument still sounds convincing to me. Plus, there's the empirical evidence that roulette betting systems don't work--just look at the opulent decor inside casinos, and the cheap/comped rooms. Casinos have money to spare.

But a year or so ago I realized that I could not prove mathematically that no possible system could work. I'm trying to fill that gap. It would be nice to be able to prove what I thought I already "knew". It is the strong law of large numbers that connects the notion of expected value to reality.

mathman,

I think you may have provided the best possible reference for my purposes. If I understood it correctly, the general result I was looking for follows from the first theorem in that paper. The result I'm looking for is that as the number of trials N --> infinity, the total winnings divided by the total amount bet (total "action") tends to p-q almost surely under these conditions: 1) It is assumed that all bets are made in units of "chips," and that there is a maximum bet limit of K chips per trial. 2) The system for determining the bet size on the nth trial can depend on the previous n-1 outcomes in any way. 3) The player bets at least the (positive) minimum bet on every round. 4) The trials are independent Bernoulli trials with probability of success p = 1-q, where p < 1/2 < q.

I'll need the corollary to the first theorem proved in the paper:

Let $$|X_n| \leq 1$$ almost surely and suppose that for all m, n

$$E[X_m X_n] = \Phi(|n-m|)$$ where $$\Phi$$ is nonnegative and

$$\sum_{n \geq 1 } \frac{\Phi(n)}{n} < \infty$$

Then the following strong law of large numbers holds:

$$\lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n \leq N} X_n = 0$$ almost surely.

To get the result I want, I'll use the following trick: The gambler's chips will be kept in two separate piles. He pays his losses from one pile. These chips have nominal value p. When he wins, the casino pays him in chips of nominal value q, which go into the other pile. This makes his expected 'nominal' value for each bet zero, but at the end we will only be interested in the absolute number of chips, forgetting their nominal values.

The random variables $\sigma_n$ indicate the outcome of trial n:

$$\sigma_n = \left \{ \begin{array}{cc}+q & \mbox{ if win } \\ -p & \mbox{ if loss } \end{array} \right$$

The number of chips wagered on trial n is $$W_n(\sigma_1, ..., \sigma_{n-1})$$, so that the 'nominal' winnings on the nth trial is

$$B_n = \sigma_n W_n$$

Although Bm and Bn are not independent, their covariance is zero because the nominal chip values were chosen to make $$E[\sigma_n] = 0$$:

$$E[B_m B_n] = E[\sigma_m W_m \sigma_n W_n] = E[\sigma_n]E[\sigma_m W_m W_n] = 0$$ for n>m

If we set Xn = (1/K) Bn, we will have a sequence of random variables satisfying the conditions of the theorem from the paper. Therefore

$$\lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n \leq N} X_n = 0$$ almost surely. Then

$$\lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n \leq N} B_n = 0$$ a. s., also.

Now $$\sum_{n \leq N} B_n = qG_N - pL_N$$ , where GN is the total number of chips gained on successful trials, and LN is the total number of chips lost on unsuccessful trials. Then the total "real" winnings in chips is $$S_N = G_N - L_N$$, while the total number of chips bet (the action) is $$A_N = G_N + L_N$$. So,

$$\sum_{n \leq N}B_n = \frac{q(S_N + A_N)}{2} - \frac{p(A_N - S_N)}{2} = \frac{S_N - (p-q)A_N}{2}$$

Putting this in the SLLN gives

$$\lim_{N \rightarrow \infty} \frac{1}{N}(S_N - (p-q)A_N) = 0$$ a. s.

Finally, since we have assumed that N <= AN <= K*N, the fraction AN/N remains finite and we get

$$\lim_{N \rightarrow \infty} \frac{S_N}{A_N} - (p-q) = 0$$ a. s., which is the result hoped for.

Does this make sense? If this is right, I like the fact that it does not even require that the bet sizes from different trials become less correlated as |n-m| increases, even though they will for most systems.

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This may be a far-out comment, but hope it is somehow helpful ( I don't know
how practical it is, tho.):

Some people have tried overcoming the house's advantage by building
a computerized physical replica/model of the roulette, and trying to
model initial conditions, etc. They would then enter the data into their
model (without the casino's knowledge), and would bet on consecutive
spaces to make up for the error.

I don't know how well it worked, but it was mentioned in the book
"Eudaemonic Pie".

This may be a far-out comment, but hope it is somehow helpful ( I don't know
how practical it is, tho.):

Some people have tried overcoming the house's advantage by building
a computerized physical replica/model of the roulette, and trying to
model initial conditions, etc. They would then enter the data into their
model (without the casino's knowledge), and would bet on consecutive
spaces to make up for the error.

I don't know how well it worked, but it was mentioned in the book
"Eudaemonic Pie".
I haven't read the book, but I have heard about people using a shoe computer and tracking the spins. Ed Thorp and Claude Shannon were the first to make it a reality, but I think it was some smart college kids who really perfected it. Seems like I remember reading somewhere (it might have been a review of Eudaemonic Pie or Fortune's Formula) that they achieved a ~20% advantage over the house, as opposed to the usual ~5% disadvantage. That's pretty sporty. In a situation that's favorable like that, you use the Kelly criterion to determine your bet sizes to get the best long-term growth rate. There's no need to fool around with one of the pattern systems then, which are what my original question applies to.