How slot machines work

How the random number generator picks the stops

by Michael Bluejay | ©2002-11

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Slot machines are random, period

Slot machines are pretty simple.  A random number generator (RNG) picks random numbers, which are then mapped to the symbols on the reels, and the machine stops on those symbols.  The selection is truly, completely random, and isn't influenced by anything at all.  There's no such thing as a "payout cycle" when a machine supposedly will pay out more than normal.  A machine is never "due" to hit.  Every spin is just like every other spin:  completely random.

That makes sense, because the whole foundation of casino gaming is randomness.  Every other game in the casino, from craps to roulette, works the same way.  The outcome is random, and the odds are simply tilted in the casino's favor.  There's no mystery about slots, just like there's no mystery about craps.  Why would there be?  The casino wants its slots to work like all its other games:  randomly.

"[A gaming device] must use a random selection process to determine the game outcome of each play of a game...Each possible permutation or combination of game elements which produce winning or losing game outcomes must be available for random selection at the initiation of each play....The selection process must not produce detectable patterns of game elements or detectable dependency upon any previous game outcome, the amount wagered, or upon the style or method of play."

There you have it.  Slots aren't affected by the presence (or absence) of a player's card, how long it's been since the last jackpot hit, or anything else.  They're random, period.

The Basics

Note that by the time the reels are spinning, the game is already over.  The RNG has already selected the stops, and the reels spin sort of as a courtesy to the player.   Slot machines don't even need reels -- you could just put your money in and the machine could tell you whether you won or lost.  The presence of the physical reels makes no difference in the game -- they're just there to show you what the computer picked.

How the stops are selected

A typical non-progressive video slot has 35 to 50 stops per reel.  A mechanical slot uses a "virtual reel" inside its programming of 64 to 256 stops, which are mapped to the 22 stops on the physical reel.  The physical reel isn't big enough to hold all the stops that are needed, so the real reel is the big one in the computer program.  (source)

If you saw a worker open up a slot machine you might see a reel like the one on the right, if it were unfolded.  There are various symbols spread across 22 stops.  Yes, the blanks count as stops.  You might think that since there are 11 blanks you have a 50% chance of hitting one, and since there's only one jackpot symbol you have a 1-in-22 chance of getting it.  But it doesn't work that way, because we're not really working with a 22-stop reel.  We're really working with an invisible reel of like 128 or so stops, controlled by the computer.  The computer will pick a number from 1-128, each of which is mapped to a specific symbol.  Here's a hypothetical map for the reel shown at right:

Say the computer picks #53.  That's a blank, and it tells the reel to stop on a blank.  If it picks #82, then it tells the reel to stop on a cherry.  If it picks #127, then the reel tops on the jackpot symbol.

Most of the numbers are for the lower-paying symbols, so that's what's more likely to get chosen. That's what we mean when we say the reel is weighted. Some symbols are more likely to be chosen than others, even if they appear the same number of times on the physical reel.

So you don't really have a 1 in 22 chance of hitting the jackpot symbol on this reel. Your odds are actually 2 in 128, or 1 in 64.

And of course, the most likely symbol is a blank. You have a 73 in 128 chance (57%) of drawing one of those.

Speaking of blanks, when the computer picks a blank, it actually picks a specific blank. Same for the other symbols that appear on the reel multiple times, like cherries and certain bars. The table above was simplified to make things easier to understand, but now that we've come this far, let's now look at how every single position on the reel might be weighted.

The fourth column (Number of Chances) shows the weighting. We've got a 2 in 128 chance of landing on the first stop (a cherry), and an 8 in 127 chance of hitting stop #5, the Red 7. Notice how the blanks surrounding the Jackpot symbol, #20 and #22, are heavily weighted. They're more likely to be selected, resulting in the "near-miss" effect. You think you just almost got the jackpot symbol, but it's really an illusion. You weren't close at all. It's like the blanks above and below the jackpot have little magnets on them.

So far we've talked about only one reel, though most slots have three, and each reel is actually weighted differently. As you go from reel to reel the weighting gets heavier, so you're more likely to hit higher paying symbols early on. By the third reel the higher-paying symbols are even less likely. This results in another kind of near-miss effect: How many times have you gotten JACKPOT, then another JACKPOT, and then... a blank? After the first two hits you're holding your breath for the third reel, but in reality your odds are poorer for getting that third jackpot symbol than they were for getting either of the first two symbols. However, for the rest of this discussion, we're going to assume that each reel is in fact identical in order to make the math easier.

A Par sheet details the probabilities for a particular machine.  Slot makers guard them religiously, but a few have made their way into the public's hands.  From them we can see that the principles are exactly as I described.  Here are the publicly-available Par sheets I know about:

• Academic report by Canadian researchers, based on various Par sheets they obtained (PDF)
• Blazing 7's slot by Bally, obtained by a slot machine collector (PDF)
• Red White & Blue, obtained by mathematician Michael Shackleford (web page)
  

Hitting the jackpot

So now that we know the weighting of the reels, we can answer that elusive question: What are the odds of hitting the jackpot? Here's the answer. Assuming we have three identical reels as listed above, then the odds of getting the jackpot symbol on any reel is 2/128. The probability of hitting the jackpot on all three reels is 2/128 x 2/128 x 2/128 = 1 in 262,144. (If you played fast at 800 spins for 8 hours a day, you'd hit the jackpot on average once every 41 days.)  This in fact is the odds of hitting the jackpot on Red White & Blue.  (See the main slot article for more on jackpot odds.)

Calculating the payback

Now that we know the weighting of the reels, we can calculate the payback for this machine, which the percentage of money the machine would pay back over an infinite number of spins.  Of course you can't play for an infinite amount of time, but the point is, the longer you play, the closer your return will come to what the payback suggests.

Our slot has the following paytable.

To find the payback of the machine, we multiply the probability of each winning hit times the payout for that hit, then add them all up, as shown in the following table. I included a "How Calculated" column if you're interested in seeing how I derived the probabilities. The numbers I use there came from the first table, above ("Total no. of symbols" column).

So this is a 96.3% machine, meaning that if you played it forever, you'd get back 96.3¢ for every $1 you put into it.  Of course you can't play it forever, and in the short-term anything can happen, but the longer you player, closer your return will come to 96.3% -- meaning you will have lost 3.7% of all the money you bet.

Of interest is that the small payouts account for most of the payback.  The single cherry alone provides nearly a third of all the money you get back from the machine.  Same for "any bar / any bar / any bar".  The jackpot itself comprises less than 1% of the total payback.

Note that some figures are not exact due to rounding.

The RNG is constantly picking numbers

The RNG is always working, even when you're not playing, picking thousands of 3-number combinations per second.  The moment you press the button or pull the lever, the RNG picks its 3 numbers for your play.  So if someone hits a jackpot on a machine you were just playing, relax, you wouldn't have gotten it had you kept playing, because you would have hit SPIN at a slightly different time than they did.  Every millisecond you delay in hitting the SPIN button results in a different combination.

The reason the machine constantly picks numbers is so that no one can discern any pattern in the number-picking process and therefore predict a winner. It's extremely unlikely that anyone could do so even if the RNG didn't keep picking random numbers all the time, because the number of random numbers in a complete cycle is astronomical, but having the RNG pick numbers all the time removes any remote possibility that anyone could predict the outcome.

Other Resources (hand-picked)

• Wizard of Odds. Nobody knows more about the math of casino gambling than the Wizard. His survey of slot returns of Las Vegas casinos was groundbreaking and legendary.
• Slot Machine Fan. Useful, no-nonsense articles about playing slot machines, without the guesses and other B.S. so common on the net.

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