by John Grochowski

Csharp random slot-machine probability-theory Updated Oct 20, 2018; C#; bnriiitb / appliedmachinelearningcourse Star 3 Code Issues Pull requests Applied Machine.

At first glance, there’s a large gap – or is it a gaping chasm? – between slot machines and table games.

You can see cards being dealt and dice being rolled, but you can’t see random numbers being generated. That leads to the impression that slots can adjust on the fly, making sure players can’t win too much in ways that can’t be done with cards, dice or roulette wheels.

Nonetheless, slots and tables share common ground that makes them more similar than players realize. Both give the house an edge by paying less than true odds on winning bets. The casino share comes from losing bets, of course, but the winners don’t quite offset the losers because they’re paid at less than the true odds.

• The probability of winning on a slot machine is 5%. If a person plays the machine 500 times, find the probability of winning 30 times. Use the normal approximation to the binomial distribution. Find the area under the normal curve over the interval (29.5,30.5) when the mean =np=500.0.05 and the standard deviation = sqrt(500.0.05.0.95).
• Identical slot machine games might be programmed with entirely different probabilities. For example, you might play a game where you have a 1 in 2000 probability of winning the top jackpot of 1000 coins, but the identical machine right next to it might only give you a 1 in 2200 probability of winning that same amount.

Not only that. Slots and table games both arrive at expected results through normal probability.

It’s that last part that gives many slot players trouble. They see that a slot game has an expected payback percentage – usually less than 90 percent on penny slots but more on higher denominations. And they hear that results are random.

Probability: Odds of Winning at Slot Machines The Odds of Hitting it Big The number of possible combinations is fairly easy to calculate. You multiply the number symbols each slot has together. Every slot machine contains a micro computer called the Random Number Generator (RNG) that is generating numbers constantly, even when the machine is not being used. The RNG typically generates numbers between 1 and several billion hundreds of times every second.

That leads to the question, “How can slot results be both programmed and random? If there are big wins, doesn’t it take cold streaks to get back to the programmed percentage? That doesn’t sound very random to me.”

Some answers:

Slot Machines Don’t Work Like That

Game designers set the odds so that normal results will lead toward an expected average payback percentage.

That’s the same situation you face on a table game.

Take roulette. On an American double-zero wheel, the game is “programmed” with 38 possible results — 1 through 36 plus 0 and 00.The numbers come up randomly, and when you win on a single number, you’re paid at 35- 1 odds, a bit less than the true odds of 37-1. That gives the house an edge of 5.26 percent. That’s the same as saying a payback percentage of 94.74 percent.

There is nothing to keep your number from coming up two or three times in a row, and nothing that says it has to come up within several dozen spins or more. But given enough trials, the random results and the odds of the game will lead to something very close to roulette’s expected percentage.

Slots work the same way, except there are thousands of possibilities instead of 38. For regular play on the reels, randomly occurring numbers are programmed, each corresponding to a reel symbol. To make up an example, the programmer might write it so that every time the random number one shows up, the reel shows a jackpot symbol; with numbers two, three or four, it shows a seven, with numbers five through nine, a triple bar, and so on. The possibilities are programmed, but when they turn up is random, just as it’s random when a 17 turns up in roulette.
After a big win, the machine doesn’t go into makeup mode. Over a long period of time, normal results according to the odds of the game will yield a normal payback percentage, and your big win fades into statistical insignificance.

Just as when a table games designer sets the rules of a card, dice or wheel game, the slot programmer sets the possible outcomes, and the pay table gives you back a little less than the true odds of hitting the winners. You can hit several winners in a row, or none, for a number of spins.

Results are random, but over hundreds of thousands of plays they will lead to something very close to the programmed payback percentage.

Cold Streaks After Jackpots Are a Myth

Probability Slot Machine Random Game

Casinos are there for the long haul, and they know that normal results in line with the odds of the game will take care of the percentages.

Imagine you’re betting three coins at a time at a dollar machine with a top jackpot of 10,000 coins, and that the machine is programmed to pay 95 percent in the long run. We hit the jackpot on our first pull. How low must the payback be over the next 999,999 spins to bring the overall percentage back to 95 percent for 1 million reel spins?

Would you believe a drop to 94.7 percent would do it? And if those 999,999 spins brought 95 percent, the overall payback including that first jackpot would be only 95.3 percent.

Normal results bring the overall payback percentage into normal range. The game doesn’t have to force any cold streaks.

That’s the same situation table players face. If I’m playing craps, betting on 12 and collect 30-1 payoffs several rolls in a row as the shooter rolls an unusual number of boxcars, does the casino have to force other numbers to come up for a while to make up for my streak? Of course not.

Someone might congratulate me on a nice win, but operators know that the 35-1 odds against rolling 12 coupled with day after day after day of play and normal probability will lead my streak to fade into statistical insignificance.

On a roulette wheel, if red numbers come up half a dozen or more times in a row and the whole table is betting with the streak, does the casino panic? No, because normal probability will lead the streak to fade into statistical insignificance and the game will pay something close to its normal percentage.

There is no mechanism to force cold streaks after big wins on tables, nor is there any such mechanism on slots. Forced results on slots are illegal in all American jurisdictions. Results have to be random, just as they are on tables.

But odds of the game can be set so that winners are less frequent than losers, and winners pay at less than true odds. Slots work that way, and so do table games. That’s enough for random results to give the house its edge.

John Grochowski has been covering casinos and casino games for nearly 40 years. He is the author of six books and his work appears in newspapers, magazines and websites around the world.

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If you’re the type of person who isn’t that great at Blackjack or Poker and feel left out when you go with your friends to a casino, you can always spend your time on the slot machines. Slot Machines are those big boxes shaped like vending machines that have bright, flashy lights and play wonky circus music. In a big casino like the ones in Macau or Las Vegas, you will have rooms with rows of slot machines and slot machines in random corners of the casino. They are the perfect gambling entertainment for the rookie gambler and the newbie casino goer. It’s a simple game and I will try to explain how it works and your chances of winning a jackpot.

All slot machines will have the same type of features. They will have:

Probability Slot Machine

• A Lever: It’s the big handle that sticks out from the side of the machine. If you pull it, you start the game.
• Reels: The spinning wheels inside the machine that have different types of symbols and pictures on them. They start turning once you pull the lever. Then all the reels start turning and stop one-by-one from left to right. You win the game based on the arrangement of the types of symbols that appear on the reels. There are generally in most slot machines three spinning reels.

Probability Slot Machine Random Picker

There are two types of slot machines that are based on how they are made. There is the classic, mechanical slot machine that turns with the reels with gears and pulleys, and the modern, computerized slot machine that uses a computer program to turn the reels. The computerized slot machines can allow casinos to be really creative, and sneaky, with how they set them up.

Let’s set up a make-believe mechanical slot machine that has three reels with 64 symbols on each. To make things simple, to get a jackpot, you need to have a ‘7’ symbol on each reel, other combinations are losers. The first reel has no effect on where the second reel stops, and the second reel has no effect on where the third reel stops, making each reel independent of the other reels.

• If each reel has 64 symbols, of which only the ‘7’ symbol can let us win, therefore the probability of getting a ‘7’ on the first reel is 1 out of 64.
• Since all the reels are identical, that means the probability of getting a ‘7’ on the second reel is also 1 out of 64.
• Same like before, the probability of getting a ‘7’ on the third reel is also 1 out of 64.
• Since each reel is an independent event, the formula to calculate the probability of getting a ‘7’ on each reel is P(Three ‘7’s) = P(‘7’ on first reel)*P(‘7’ on second reel)*P(‘7’ on third reel)
• Therefore: P(Three ‘7’s or Jackpot) = (1/64)(1/64)(1/64) = 1/262144 or 1/(2^18)
• The odds of getting a jackpot or three ‘7’s in a row are: P(Jackpot)/P(Losing) = P(Jackpot)=P'(Jackpot) = P(Jackpot)=(1-P(Jackpot)) = (1/262144)/(262143/262144) = 1:262143. If you really think about it, it seems almost impossible to win.

People sometimes think that some machines are hotter than others in the way that they are more likely to give a jackpot because more people have played them before, when in fact it is not true at all! Because the reels are independent events and each lever pull is a new event every time, your odds of winning a jackpot will remain 1:262143, even if you play for 12 hours straight non-stop.

The Computerized Slot Machines are harder to calculate because they use complex computer programs to generate the random numbers for which symbol appears on the machine. I won’t be able to give you a calculation, but I will try to explain how casinos use probability to ruin our chances of winning:

• Let’s say we use a 3-reeled machine that has 21 symbols this time on each one.
• First the lever is pulled
• The computer generates three random numbers. For example: 12,34,56. The first number tells which symbol will appear on the first reel and so on.
• If you noticed, we have 21 symbols, so why are there numbers like 34 or 56? This is because there actually 4 reels, 3 mechanical and 1 virtual, in the matrix…
• The virtual reel has 64 numbers from 1 to 64 (could also be 32 or 128 and so on). Each of these 64 numbers match to one of the 21 symbols in each reel.
• But that would mean some of the 21 symbols will have more numbers from the virtual reel than others. Exactly, that means so symbols are more likely to appear than others. So your chance of getting a jackpot number in one reel is not 1 out of 21, but even less.
• What casinos will do is have the symbols that appear before and after the jackpot symbol on each reel to be those symbols with more numbers on the virtual reel. It gives the false idea to players like us that we are so close to winning, and so we need to play more until we get a Jackpot, but in reality you’re being a tricked and are losing a lot of money.

So next time your friends abandon you at the bar while they make millions, or lose them, at the Blackjack or Poker table, you can entertain yourself with a slot machine, but will be more prepared to stop if you think you’re close to winning after 20 pulls.

(some random dude on Google :P)