The maths behind slot machines


Maths. Remember, that subject you hated at school? Well, it’s actually kind of important in the world of slot machines and casinos. Who’d have thought it?

Here we will go into detail about the maths behind slot machines. Every gaming machine is designed to pay the player back a percentage of what is played. Fact. The amount paid back varies from machine to machine and from casino to casino. There is one thing that all machines do have in common, though. The longer a machine is played, the closer the payouts will be to the theoretical results.

The maths behind the slots can take time to understand

Slot machines use a random selection process to achieve a set of theoretical odds. Random selection, however, means that each and every time the lever is pulled and the reels are set in motion, a combination of symbols upon the reels are randomly selected. The ‘random’ part of this comes in the way that each pull of the handle is independent from every other pull. This means, the results of the previous pull, and the one before that (and the however many before that too) have no effect on the current one.

Theoretical odds are built into the design and program of every machine. It is actually possible to calculate the exact payout percentage for any machine in the long term because of this.

Other than video slots, slot machines have several wheels (known as reels) with symbols printed on each wheel. Each reel symbol represents a stop, which may come to rest on the payline. Where these symbols stop may or may not be part of a combination of symbols, resulting in a payout. The likelihood of winning any payout on any slot machine is related to the number of reels and the number of symbols on each reel.

Looking at the most popular slot

The most common type of mechanical slot machine has three reels, with twenty symbols on each reel. Quite simply, to calculate the total number of possible combinations of symbols on a machine like this, you multiply the number of symbols on each reel by the number of stops on each of the remaining reels. For the above examples, this would be 20 x 20 x 20 = 8,000 combinations of slot symbols.

Let’s say the jackpot offered on this machine pays on the symbols 7 7 7 appearing. For argument’s sake, let’s say that there is only one 7 symbol on each reel. Therefore, the probability of hitting this jackpot is 1/20 x 1/20 x 1/20 or more simply put: one in 8,000. Still with me? No? Should have listened more in maths class then…

As with the above example, it’s possible to calculate the probability of any combination of symbols hitting if we know the number of times each symbol appears on each reel.

Even though the maths can be a little complicated sometimes, the theory behind it is simple. Every slot machine has calculations behind its possible outcomes.

With technology comes difficulty

There are many variations when it comes the the reels and combinations of each slot
There are many variations when it comes the the reels and combinations of each slot

Back in the day, when mechanical slots dominated, it wasn’t too difficult to count the symbols on each reel and determine exactly the payout of a given machine. It could and did happen. Now, however, with microprocessor controlled slots, this task is pretty much impossible. The number of stops per reel can be as many as 256. To determine the payouts of a machine like this would require seriously significant reverse engineering and is beyond the scope of almost any player. Yep, even you.

The number of reels on a machine has a greater effect on the probabilities than the number of symbols per reel does. Remember that! It’s important.

Compare a machine with 32 stops per reel and three reels with a 22 stop per reel machine and four reels, you will see the tremendous difference another reel makes:

• 32 Stop, 3 Reel: 32 x 32 x 32 = 32,768 combinations

• 22 stop, 4 Reel: 22 x 22 x 22 x 22 = 234,256 combinations

If we consider a five reel machine with 32 stops per reel, there is over 33 million combinations!

Each slot machine has a predominated payout percentages. For example, if a slot machine advertises that it will “pay back 97.8%” then what this really means is that for every dollar inserted into said machine, it will return 97.8 cents. So, how about thinking of it this way?
For each dollar put into that slot machine, the casino keeps 2.2 cents. Just blew your mind, right? Remember, these percentages only hold true over very long-term play consisting of thousands (or even millions) of plays!

People often misinterpret the above percentages and think it means that if they play with, for example, $100 on a 97.8% payback machine then they can only lose $2.20. This line of thinking doesn’t quite work out. Firstly, theoretical percentages will only ever be attained over long periods of play. Over a few dozen, or even few hundred rolls, the payback percentage varies much greater. Secondly, if a player did bring along $100 for slot play, they would not usually limit their play to inserting this amount of money at one time. Generally, players use smaller amounts and then top up when necessary, continuing this patter until they are out of the money they wanted to play with. The percentage theory doesn’t quite work this way.

The reality is, a casino extracts its percentage on every coin inserted into a machine. The player cannot limit their play and the machine will (over the long term) continue to grind away at all the money played. This is the reality of the maths behind the slots.

Table 1 (below) shows the devastating effect the house edge can have on the player’s bankroll. Table 1 compares slot hold percentages from 2% to 15% for ten rounds of play, starting with $100.

Table 1. Amount Retained Per Round of Play

Slot Hold % 2% 5% 10% 15%

Start Round $100 $100 $100 $100
1 98 95 90 85
2 96 90 81 72
3 94 86 73 61
4 92 81 66 52
5 90 77 59 44
6 88 73 53 38
7 86 69 48 32
8 85 66 43 27
9 83 63 39 23
10 81 60 35 20

The table shows that with a 15% casino hold, there is only $85 left after one round of play. After ten rounds, the $100 has been reduced to only $20. Contrast this with the 2% hold rate and you can see that even after ten rounds, you would still retain $81. Even though you would have gradually lost some money on this machine, hitting a single higher payoff would put you ahead and you would also have gained much more playing time to do so. Always seek the machines with lower hold percentages!

Unfortunately, casinos don’t label their machines with the hold or payback percentages. (I wonder why?!) However, the player’s win rates are available for different locations. Do your research and you can find this information out. Highly advisable!

A little bit of info goes a long way

Doing your research beforehand can increase your chances of winning.

Examining an individual slot machine in a little bit of detail can further show how slots are programmed to pay off. Table 2 (below) examines a three reel, two coin multiplier which pays bonuses off on two of its payoffs. The below tables shows the pay schedule for this machine.

Table 2. Pay Schedule for Option 3 Reel 2 Coin Multiplier

Symbols 1 Coin 2 Coins

7B 7B 7B 200 1000

5B 5B 5B 50 150

1B 1B 1B 10 20

AB AB AB 5 10

— — — 1 2


7B = Seven Bar

1B = One Bar

5B = Five Bar

— = Blank or ‘Ghost’

Bonuses paid on 7B 7B 7B and 5B 5B 5B when two coins are played.

The first step in properly analyzing this machine is by breaking out the number of symbols (or stops) per reel. The machine in question has 32 stops per reel and the reel analysis is shown below in Table 3.

Table 3. Reel Analysis Option 3 Reel 2 Coin Multiplier

Number of Symbols per Reel

Symbol Reel 1 Reel 2 Reel 3

7B 2 1 1

5B 5 4 4

1B 9 9 9

— 16 18 18

To explain this requires the analysis of a few numbers (we’ve warned you). With three reels of 32 symbols each, there is a total of 32,768(!) combinations of symbols possible. This is 32 x 32 x 32. Since bonuses are offered when the second coin is played, that adds another 32,768 combinations with the play of the second coin. Therefore, one coin being played = 32,768 combinations. Two coins = 65,536. Double the amount. Now, Table 4 shows an analysis of all winning combinations on this machine.

Table 4. Analysis of Winning Payoffs

Combination # on Reels Hits Deduct 1 Coin 2 Coins Payout%
7B 7B 7B 2 1 1 2 -0- 400 2,000 1.4%

5B 5B 5B 5 4 4 80 -0- 4,00 12,000 14.0%

1B 1B 1B 9 9 9 729 -0- 7,290 14,580 25.6%

AB AB AB 16 18 18 3,136 811 11,625 23,250 40.8%

— — — 16 18 18 5,184 -0- 5,184 10,368 18.2%

Totals 9,131 811 28,499 62,198 100.0

Less Deducts – 811

Net Hits 8,320

Looks complicated, right? So, let’s explain.

The first column shows each winning combination. In the second column are the numbers of symbols on each reel. For example, for the combination 5B 5B 5B, there are five 5B symbols on the first reel, four on the second reel and four on the third reel. The next column then shows the total number of winning combinations (known as Hits). For the above 5B 5B 5B combination, we have 5 x 4 x 4, which = 80 hits. The next column ‘Deduct’ shows the number of times that a symbol is used to compute the different payoffs, with the same symbol used. It’s deducted so that the same symbol isn’t deducted twice.

Those of you with the eagle-eyes will be asking why are 811 hits deducted from the total number of hits for the AB AB AB combination? This is because 729 of the Bar symbols will be the combination 1B 1B 1B and 80 of the Bar symbols consist of the combination 5B 5B 5B for a total of 811.

The Payout columns are broken down in payouts for one and two coins played. The amount in these columns have been computed by multiplying the payoff for each combination of symbols times the number of Hits for that combination.

Returning to the 5B 5B 5B combination, the payouts are computed for one coin as 80 Hits x 50 coins for a payout of 4,000 with one coin played. When two coins are played and the same combination shows, the payouts are computed as 80 Hits x 150 coins which = 1,200 coins, reflecting the bonus payoff.

Add up the total number of hits and you get a total of 9,131, before deducting overlapping symbols. Deducting 811 for overlaps gives a net total of 8,320 hits that will pay off on this machine.

To compute the amount the slot will retain, divide the total number of payouts by the total number of possible combinations for:

Payouts % of Payouts Total Combinations Payout Percent
1 Coin 28,499 32,768 86.97%
2 Coins 62,198 65,536 94.90%

Look at the last column of Table 4 and you’ll notice that almost 85% of the payouts occur on the lower paying combinations. This is an important fact for a winning strategy.

All of this adds up to how the maths behind a slot machine works. It isn’t simple. It isn’t easy. But, it is vital to know the facts behind the game when you’re developing a strategy. As always, the more research done, the more chance you have.