Today, the mathematics of slot machines. The University of Houston mathematics department presents this program about the machines that make our civilization run, and the people whose ingenuity created them.

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Mathematicians first got interested in randomness by studying games of chance. Ever since, the histories of mathematics and gambling have been intertwined. Clever gamblers use mathematics to look for the smallest advantages, and casinos use sophisticated mathematical tools to devise new ways of drawing in players.

Indeed, a patent granted to the Norwegian mathematician Inge Telnaes in 1984 transformed the gambling industry. Prior to Telnaes’ invention, slot machines were essentially mechanical devices. Besides being difficult to tune and maintain, mechanical slot machines suffered from an essential problem: Let’s look at a machine with three reels, each with 12 symbols, with one of those 12 symbols a cherry. The likelihood of getting three cherries, and winning the jackpot, is 1 in 1,728. If the casino wants to make money, the jackpot payout should be, say $1,700 on a $1 bet. That does not seem attractive by today’s standards. However, the only way to increase the payout is to decrease the chances of hitting a jackpot.

Adding another reel is a possibility. For instance adding a fourth reel in the previous example would get us to a jackpot of about $20,000. But people do not like machines with more reels — they intuitively, and rightfully, feel that extra reels diminish their chance of winning. Another possibility is to put more symbols on each reel. But the astronomical jackpots you see in casinos today would then require truly enormous machines.

Inge Telnaes proposed a simple solution: Let a random number generator — a computer chip — determine the combination of symbols that appear when the reels stop. In other words, use a chip to control where the reels stop on a spin, but create the illusion that the wheels stopped on their own. The number of possible outcomes on the slot machine does not change. However, by reprogramming the chip, the operator has full control over the likelihood of each of the different outcomes. For instance, the operator could make the three cherries appear only once in a million spins.

This was a brilliant insight: Suppose I pick a number between one and a million. Would you be willing to bet that you can guess that number? The answer is probably not. But let a computer chip pick such a number, put the chip in a machine with blinking lights and spinning reels, and many people will be more than willing to make the bet. It is simply because what people assume is happening in a slot machine is very different from what is actually happening.

The Magician oil painting by Hieronymus Boschfrom between 1475 and 1480

The history of gambling is also intertwined with that of a less reputable group — tricksters and swindlers. In the long run, the only sure way to make money by gambling is to create the illusion that your opponent can win, while keeping the odds firmly on your side. And that gives those who know math a very solid advantage.

I'm Krešimir Josić, at the University of Houston, where we're interested in the way inventive minds work.

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NOTE: In the example with three cherries, I assumed that one only wins in the case the spin results in three cherries, and there is no other winning combination. In actuality, there are typically many winning combination, and as a result, the jackpot would have to be even smaller.

The following story in Wired Magazine shows the drawbacks of the new generation of slot machines — they are easier to hack and to counterfit than their mechanical counterpart http://www.wired.com/magazine/2011/07/ff_scammingslots/.

Here is a more exhaustive discussion of the history of slot machines, and the random number generators within them http://catlin.casinocitytimes.com/article/non-random-randomness-part-1-1243. You may want to scroll towards the end of the article to read about how flaws in the design of gambling machine resulted in somebody picking 19 out of 20 winning numbers in a game of KENO — and doing so 3 times in a row. That person walked away with $620,000, but only after some controversy.

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Both images are from Wikipedia. The slot machine image was taken by Jeff Kubina.

For more mathematics in everyday life, visitkjosic.wordpress.com.

This episode was first aired on September 7th, 2011

The house edge at slot machines isn't based on how lucky the players are, it is based on the pre-programmed electronic software used. And, that house edge varies from casino to casino and from geographic location and state to state.

Don't be fooled by a billboard that says 'Our slots pay 99%' because that only pertains to a small number of machines, not the casino's overall house edge. All video slot machines (including poker) use a Random Number Generator to provide accurate odds for the player and the house.

The number of winning hands (symbols) in any program is mathematically proven before the slots hit the gaming floor, and while the odds vary for each player (because this really is gambling), overall, the anticipated house edge will eventually be achieved.

The House Edge at Slots

Each gaming jurisdiction requires all slot machines to be registered and all slot manufacturers must submit their programming and math charts to the state gaming office, such as the Nevada Gaming Control Board. While the state may only require a minimum payback of 86-percent, most video poker and slot machines pay more. The average payback in Nevada is lowest on penny slots at about 90-percent. Dollar slots and some Deuces-Wild poker machines have the highest payback at nearly 99-percent.

Of course just because a machine has a payback of 99 percent does not mean you will win back $99 for every $100 you play through the machine. That payback amount is an average. Over the course of an evening of gambling, you might lose $500 or win $5,000. That's the beauty of the gamble, right? And, the casino doesn't care either way, because they win an average of whatever their house edge is over the course of the month.

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How does that 99-percent work for you? Aside from the ups and downs of jackpots and dry spells, your entertainment dollar is going to be based on how long you play. If you play a machine that takes $3 per spin and you pull the handle 10-times an hour, you are playing $3 x 10 x 60 = $1,800 in action. At 99-percent house edge, your play costs $18 an hour. Some nights you'll go home ready to kick the wall, and other nights you be singing a happy tune because you hit a jackpot and skated out with hundreds. That's gambling.

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Play Within Your Means

As a general rule, the higher denomination slot machines have a greater overall payback percentage that the lower ones. That's great for those with a healthy bankroll, but you'll need to play the games that offer you the best bang for the buck. The worst thing a player can do is to play any game for more money than they can afford to lose or at a higher denomination than their bankroll will handle.

Another consideration is making sure you can qualify for any bonus screen or Royal Flush payoff. If you want to wile away the hours playing a single nickel in a machine, that's great. Enjoy. You won't win or lose much and you'll probably go home happier than most players. But your play comes with a catch!

That catch is that the house edge provided for each machine includes the big jackpots (such as a Royal Flush), and if you play below the minimum to reach the bonus screen, the house edge is much higher! You won't lose much, but you'll never win more than a couple of dollars.

The Best Option

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The best option is to play the games you enjoy most, and make sure you read the Help Screen so you know how many credits it takes to qualify for at least the lowest progressive jackpot and the bonus screen (or Royal Flush at video poker). As a recreational player, your first goal is to have fun. If you only visit the casino occasionally the overall payback of the machine won't make much of a difference. If you visit regularly, make sure you join the Player's Club and play within your means!