Are RNGs really random?

Dear John,

Thank you for your contribution at presentation of the truth about slot machines.

"Your homework is to read articles about Random Sampling with Replacement and virtual reels."

I did that homework many years ago at University when I studied the Probability. But, at least you highlighted a key point of our disagreement.

The probability with replacement is not applicable on "RNG patterns." Period. To calculate the probability with replacement you need to know exact structure of the domain where are you taking samples from. You have 100 randomly picked numbers. You are pulling just one number. What is probability that you will pick up number 12345? You do not know if that number exists in the domain at all or how many times repeats in the domain. How can you calculate the probability?

"A machine's long-term payback is calculated using the pay table and the layout of the symbols on the virtual reels. The RNG is irrelevant."

Your statement is mathematically incorrect. The probability with replacement is applicable on the structure of either the real reels or virtual reels, excluding RNG pattern. And yes, you can calculate it based on an actual pay table. But probability calculated on that way is irrelevant, due to the fact that final outcome or final probability is DICTATED by the RNG pattern. Nobody can calculate final probability, even he knows exact "RNG pattern," due to the cycling and timing. Period.

"The games are fair, however, as compared with carnival games, which use optical illusions and other tricks to make the games appear easier to win than they actually are."

The Telnaes style slot machines or slot machines with virtual reels are optical illusions and tricks. The player is under impression that he is playing with 22 stops, but no: he is actually playing with 32 or more virtual stops! He sees three 7s on one reel and he is tricked because two of them will never appear on the payline! Is it a fraud?

More in the next e-mail.

Best regards,
Mike

I asked Mike to define exactly what he meant by "RNG pattern." His reply follows.

Obviously I am not good at presenting my statement in a very efficient way.

The "RNG pattern" includes: the RNG algorithm, hardware and way of taking the samples. We agree that the true random number generation is not implemented in the slot machines. You call it pseudo RNG.

Pseudo RNG is based on the mathematical formula/algorithm and the seed. The seed is starting number.

As an example let's take most common pseudo RNG function: Xn+1= C*Xn mod P, X0=seed and C is a constant

Let say P=100 and X0=54 and C=263

Let's generate several numbers:
X2=263*54 mod 100=2
X3=263*2 mod 100=26
X4=263*26 mod 100=38
X5=263*38 mod 100=94
X6=263*94 mod 100=22
X7=263*22 mod 100=86

We can generate pseudo-random numbers as many as we want. Let's say we have generated 1,000 pseudo-random numbers. Those 1,000 numbers are not present all the time in the machine. If we are talking about a three reel slot machine, we have a three memory locations for the numbers. When you press the Stop push-button you have only three numbers available! Let's say we pulled X4 as first number. X5 is dependable on X4 and X6 is dependable on X5. So, the pulls are not independent. These two facts are breaking established theory of probability with replacement. The probability clearly said that all numbers have to be in the pool and each draw has to be independent.

That is not happening with computerized slot machines due to the hardware limitations.

All numbers do not have equal probability to appear at the moment when you press the Stop pushbutton. So, the probability either with replacement or without replacement is not applicable.

I am open for further discussion.

Best regards,
Mike

Dear Mike,

Thanks for the clarification. It helps me see exactly where we disagree.

There are some who don't believe that, given an unbiased selection mechanism, as you take more and more samples, the characteristics of your sample nears the characteristics of the population with Random Sampling with Replacement. Hence your homework assignment to read the literature to see that that is true.

I see now though that our area of disagreement is whether the RNG is unbiased. I suspect that this is a point on which we will have to agree to disagree.

Let's imagine that you will take part in a Turing Test. Your goal is not to tell machine from human, but rather to tell pseudo-random number generator from true random number generator. You are presented with two number streams, one from the P-RNG and the other from the true RNG. You have to identify the source of each number stream.

There are many statistical tests you can apply against the number streams to help you decide. You can find them by searching on the Internet. You can also find a list of the statistical tests that Gaming Labs International uses to test P-RNGs in its GLI 11 - Gaming Devices in Casinos V2.0 document (section 4.3.3 Applied Tests).

The P-RNG mathematicians and programmers have designed their P-RNGs so they pass those statistical tests for randomness. You would not be able to tell the number stream from the P-RNG from that from the true RNG.

Now, I know you don't have access to slot machine performance reports, but if you did you would see that as machines get more and more play, their actual payback percentages get nearer and nearer to their long-term payback percentages (percentages that are calculated using the pay table and the layout of the symbols on the virtual reels). If the P-RNGs were biased, the actual percentages would be homing in on some other numbers than the machines' long-term paybacks.

Moreover, each slot machine has a PAR (or PC) sheet that details the virtual reel layouts and specifies the machine's long-term payback percentage. I've always been able to verify the long-term payback on the sheet by using the virtual reel layouts and pay table. Gaming jurisdictions want to ensure that machines are operating properly and that they are receiving all of the tax revenue to which they are entitled. The long-term payback on the PAR sheet is the metric against which a machine's performance is measured.

I agree that if the P-RNGs were biased, we would need to know the true probabilities of choosing each virtual stop in order to calculate a machine's long-term payback.

But I disagree that the P-RNGs are biased. On any given spin, each virtual stop on each virtual reel has the same probability of being chosen. Testing labs like GLI test the number streams from the P-RNGs to ensure that this is true.

As I said before, we'll have to agree to disagree about whether P-RNGs are biased.

Best of luck in and out of the casinos,
John

Send your slot and video poker questions to John Robison, Slot Expert, at slotexpert@comcast.net. Because of the volume of mail I receive, I regret that I can't reply to every question.