Diamond Gambling For Random Equipment
By John Robison
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One of the questions I'm asked most often is - 'how is it possible for a slot machine to be a random device and for a machine to also pay back a certain percentage of the money played through it?' If the results are truly random, people argue, then the payback should be random too.
Despite the fact that this governor function does not exist and in all jurisdictions whose regulations I’m familiar with, it is possible for the results on slot machines to be determined at random and for machines to have specific payback percentages.
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It’s difficult to reconcile the fact that results on slot machines are random and for us to be able to know what will happen overall on a machine. Let me illustrate how this is possible with my RWB Ping Pong Ball Game.
Suppose I have a basket that contains 100 ping pong balls. Eighty ping pong balls are white, 15 are blue, and five are red. You draw a ball at random from the basket. There’s a cover on the basket, so you can’t tell what color ball you’re drawing. Also, you can’t tell the balls apart, so you’re no more likely to draw one ball over any other. After you draw the ball, you record the color and put the ball back in the basket.
As you repeat this action, you will find that the percentage of draws that were red balls gets closer and closer to 80%, the percentage of draws that were blue balls gets closer and closer to 15%, and the percentage of draws that were red balls gets closer and closer to 5%.
We know that this will happen because you draw the balls at random and each ball is equally likely to be drawn by you. But it is not equally likely that you will draw any particular color. Eighty percent of the balls are white, so we expect 80% of your draws to be white. Similarly for the blue and red balls.
Even though you drew the balls completely at random, the distribution of colors you recorded will match the distribution of colors in the total population of ping pong balls in the basket. Random does not mean that everything is completely unpredictable and unknowable.
Now let’s make the game more interesting. You have to pay me $1 each time you want to draw. When you draw a white ball, I keep the dollar. When you draw a blue ball, I return your dollar. And when you draw a red ball, I pay you $16. A nice payoff for drawing the red ball.
Looking at this game from my perspective, I have a 15% chance of paying you $1 and a 5% chance of paying you $16. Calculating this out, we have 0.15(1) + 0.05(16) = 0.15 + 0.80 = 0.95. On the average, then, for every $1 you give me to play, I will return 95 cents to you.
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The RWB Ping Pong Ball Game is just like a 95% payback slot machine. Even though the outcomes in both games are chosen completely at random, each pays back 95% of the money played in the long run.
A slot machine works very much like the RWB Ping Pong Ball Game. Conceptually, there is a basket of ping pong balls for each reel. But instead of having different colors on the balls, these balls have symbols representing the different symbols on the reels on them. Some symbols appear on more balls than other symbols.
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Let’s set up a basket of ping pong balls for a reel from a real Double Diamond slot machine. Even though there are 22 symbols and blanks on the reel, our basket will have 72 ping pong balls in it. Having more balls in the basket than we have stops on the reel allows us to alter the probability of landing any particular symbol on the payline from what it appears to be from counting the number of times it appears on the reel and dividing by 22.
The reel has 11 blanks on it, but I’m going to put 31 blank ping pong balls in the basket. Half the stops on the physical reel are blanks, but slightly less than half the ping pong balls are blank, so it’s actually a little less likely than it appears for a blank to land on the payline on this reel. There’s only one cherry on the reel, but I’m going to put two ping pong balls with cherries in the basket.
All of the other symbols – single bar, double bars, triple bars, 7, and Double Diamond – appear twice on the reel, but the number of ping pong balls with each symbol in the basket is 25, 4, 6, 2, and 2, respectively. We need about 6.5 ping pong balls with each of these symbols in the basket to have the probability of drawing a ping pong ball carrying the symbol be the same as what appears to be the probability of having the symbol land on the payline from looking at the reel, so you can see how different the true probabilities are from what the reel makes you think they are.
Just as with the RWB Ping Pong Ball Game, the more you play this Double Diamond machine, the closer the distribution of symbols landing on the payline will get to the distribution of ping pong balls with each symbol in the basket.
Even though it sounds like random should mean completely unpredictable, the only thing we can't predict is what color ping pong ball you'll draw next or what symbols will appear on the payline next. Because we know the distribution of ping pong balls in our basket and the casino knows the distribution of symbols on the slot machine's reels, we can predict -- no, more than predict, we can calculate with near certainty -- how much my game and how much a slot machine will pay back in the long run.
The only reason we can't be dead certain of how much a machine will actually pay back is because the outcome of each game is truly chosen at random and there is no function forcing the payback to a particular number. But given enough draws or spins, my RWB Ping Pong Ball game and a 95% payback slot machine will both pay back very, very, very close to 95%.
To sum up, there's no attempt in the programming of a machine to force it to a particular payback percentage. There’s no need to. Random sampling from a known population takes care of it automatically.
Send your slot and video poker questions to John Robison, Slot Expert™, at slotexpert@slotexpert.com.
Because of the volume of mail he receives, John regrets that he can't reply to every question.