The Math Of Slot Machines

This is the also the basic idea of my crazy Moonshot slot strategy. I attribute the incredible popularity of the Cleopatra slot to the combination of a very popular theme, balanced math and hypnotizing sounds and voices. The game was truly one of the best slots at the time of its release to land based casinos. Coin-in, simply speaking, is the total amount of bets made. Let’s say you pick a slot machine and bet $1 per spin and do 100 spins. Your total coin-in is $100. Coin-in is not a count of how much currency you’ve inserted into the machine, but instead a cumulative total of wagers made.

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The 16-reel slot machines usually have the 4 x 4 configuration of the display. The standard length of a payline is 4, but it could also have the length 3, 6, 7, or 8. The 16-reel 4 x 4-display slot machine could have 8 to 22 paylines of length 4, as follows: 4 horizontal, 4 vertical, 2 oblique (diagonal), or 12 trapezoidal lines. It could also have 4 transversal stair lines of length 7, 12 double-stair lines of length 6, or 10 double-stair lines of length 8.

The Math Of Slot Machines

Slot machines use maths to operate, so why can’t you use mathematics too beat them? Let’s fight fire with fire. Find below a math based approach to playing online slots at sites such as Thorslots.com.

Understanding the mathematics behind online slot games is invaluable in helping players choose the best machine. So let’s start by explaining how mathematics is used to operate slot machines.

Slots mathematics

The outcome of each spin on a slot machine is determined by a Random Number Generator (RNG). These Random Number Generators use mathematical formulas to generate random numbers, which correspond to the symbols on any given slot. If the random numbers generated correspond to a winning symbol combination, you win.

Calculating probabilities playing slots games

Probability refers to the likelihood of players winning at a slot machine. In order to calculate the probability of a specific slot, you need to know what a specific winning combination pays, as well as the probability of landing that combination.

How much any given combination pays out is usually listed in the paytable, but how do you calculate the probability of landing that combination? Easy. Just calculate the amount of stops in that slot. Let’s use a simple example. Let’s say that a slot contains 3 reels, with 5 symbols on each reel. To calculate the number of possible combinations on this machine, just multiply the number of symbols on each reel, by the number of stops on each reel. So this would be 5 x 5 x 5 = 125 possible winning combinations. If the jackpot in this slot pays out only for three cherries across the reels, and only one cherry occurs per reel, the calculation would be as follows: 1/5 x 1/5 x1/5 which equals a 0.008% chance at hitting the jackpot.

It is thus possible to calculate the probability of landing any combination as long as you know the number of times that each symbol appears on every reel, which is not information usually shared by online slots. Modern machines also come with many different numbers of reels and winning combinations, which could make it difficult to calculate its probability.

It is quite possible to calculate the probabilities of a simple and traditional slot machine. However, with modern slots it would be pretty much impossible, as slots can have as many as ten reels and 200 stops.

RTP percentages for online slot games

Return to Player percentages (RTP) refers to the overall percentage of total wagers that a video slot game will pay out back to players over a period of time. Each online slot has a Return to Player percentage. For example, if a slot has an RTP of 98%, that means that for every dollar you bet, you will get back 98 cents. So to increase you odds of winning, you would want to choose the slot machines with the highest RTP rates, preferably 90+.
These are just a few of the ways in which you can use maths to choose the right online slot and improve your chances at winning.

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.

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.

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

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The Math Of Slot Machines

The Math Behind Slot Machines

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.

The Math Of Slot Machines Games

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

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

This episode was first aired on September 7th, 2011


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