N Slot Gambling Machine Expected Value

When you read the seemingly endless stream of slot machine jackpot stories churned out by the media, it can be easy to trick yourself into thinking the old one-armed bandits can be beaten.

As far as the casino industry is concerned, slot machine games are the most played games, yet, they are the least understood. In the UK, slot machine games, also known as the fruit machine games, are some of the most sought after games at the land based casinos, and in the US, where they were first played in the 1890’s, they continue enjoying prominence and popularity among the players. Expected Value at the Casino. Now turn to the casino. In the same way as before we can calculate the expected value of games of chance such as roulette. A roulette wheel has 38 numbered slots from 1 to 36, 0 and 00. Half of the 1-36 are red, half are black. Both 0 and 00 are green. Suppose you start with $100 and bet a dollar at a time, for example. After inserting all $100 into the slot, 100 pulls later you’ll end up on average with $90, because you lose 10 percent of your money. If you run the $90 back through the machine, you’ll end up with 90 percent of it back, which is 0.90 x 90 = $81.

After all, if an anonymous player with only $40 deposited in a Megabucks machine can walk away with a $13.1 million jackpot, why can’t the next multimillionaire to beat the odds be you?

Well, it certainly could be, but making that happen is almost entirely reliant on random chance. Slots just aren’t like their skill-based casino counterparts like blackjack and video poker. You can’t play better than other gamblers or use skill and savvy to even the score versus the house.

You don’t get to make decisions that influence where the reels will come to rest, and lacking those essential elements of a skill game, the slots become pure games of chance.

With that said, some slot players out there definitely play worse than others. By failing to recognize the crucial connection between payback percentage rates and expected value or by believing in B.S. systems that claim to make every slot session a sure thing, millions of poor slot players combine to dump billions of dollars into the casino industry’s coffers every year.

You don’t have to be one of them, though, thanks to the six ways to become a better slot machine player listed below.

By adding these tools to your arsenal of slot strategies, you’ll always bet against the best possible odds while avoiding the sucker machines that gobble up bankrolls before less informed players even know what happened.

1 – Don’t Fall for Myths, Misconceptions, Superstition, or Straight-Up Snake-Oil

Spend enough time around the slot machine parlor, and you’ll invariably hear fellow players discussing their preferred systems and strategies.

If they’re talking about payback percentage comparisons, game selection, and bankroll management, by all means, let their wisdom wash over you. Those are thinking players who use common sense and logic — but they’re also a rare breed.

More often than not, when you hear a slot spinner start talking about the “secrets” to winning consistently, they’re referring to completely nonsensical myths and misconceptions.

Some players just refuse to believe that a gambling game can be based on chance alone, so they resort to illogical ideas like “hot” and “cold” machines, casino supervisors controlling when and where jackpots are paid, and even codes appearing on the reels that are indecipherable unless you know what to look for.

The most commonly used slot systems tend to be based on erroneous beliefs, so don’t be surprised to see regulars getting up every few minutes or so to try a new machine.

According to these superstitious players, if a slot isn’t landing on certain symbols right out of the gate, it’s a “cold” game that won’t produce a jackpot for days or months.

Other players walk the floor like scavengers, lying in wait before snagging a machine after it’s been “primed” by an unlucky tourist. These slot vultures believe that a machine is more likely to land a jackpot combination only after it has claimed a certain number of coins.

You’ll even see some players staring closely at the reel screens after every spin, searching for inscrutable patterns like “zig-zags” and “Vs” linking special symbols. For these players, the appearance of a pattern on the reel screen signals that a slot is ready to dispense its most lucrative topline payouts.

Of course, none of these “foolproof” systems for beating the slots will actually work.

No matter what many slot players try to tell themselves, every single spin you’ll ever play represents a wholly independent event.

Previous spins can’t affect the future, so a machine is just as likely to trigger a jackpot on weekdays as opposed to weekends or when it’s situated in high-traffic areas or off in the corner.

The probabilities have been preprogrammed into the machine’s random number generator (RNG), so there’s no amount of pattern-tracking that can help predict when jackpots will be awarded.

You wouldn’t know that from researching slot strategies online, though — not with so many snake-oil salesmen out there trying to trick hopeful gamblers into spending big bucks on utterly useless products.

You don’t want to be the sucker these hucksters count on to line their pockets, so head here to read up on game selection, payback shopping, and other slot machine strategies that really work.

2 – Only Bring Your Bankroll to the Best Slot Locations Around

By now, most players know that the corporate-owned casinos on the Strip offer the worst slot odds in Sin City.

But until you see the data spelled out clearly, it can be difficult to gauge just how awful Strip slots really are. Fortunately, the Nevada Gaming Control Board (NGCB) performs annual surveys to compare win rates (for the casino) to player payouts. This data allows the NGCB to calculate average payback percentage rates for all slots in the Silver State — rates which can be broken down based on region and coin denomination.

The Strip – 88.38%
Downtown – 89.15%
Boulder Strip – 90.38%
Las Vegas – 90.88%

The Strip – 91.64%
Downtown – 93.40%
Las Vegas – 95.29%
Boulder Strip – 96.21%

The Strip – 89.35%
Downtown – 94.25%
Las Vegas – 96.41%
Boulder Strip – 96.13%

The Strip – 92.43%
Downtown – 94.63%
Las Vegas – 95.39%
Boulder Strip – 96.13%

The Strip – 87.83%
Las Vegas – 88.52%
Downtown – 89.07%
Boulder Strip – 91.35%

The Strip – 91.96%
Downtown – 92.67%
Las Vegas – 93.53%
Boulder Strip – 94.34%

As you can clearly see, warnings about the Strip are far from superstitious — they’re entirely accurate. Across the board, you’ll always face the worst odds when spinning on the Strip, so take your bankroll to Boulder City, Downtown, or North Las Vegas instead.

3 – Choose a Higher Coin Denomination to Increase Your Payback Percentage

Take a second look at the data shown above, but focus on the coin denominations rather than the region.

When you do, you’ll notice a close correlation between bigger bets and a better payback percentage for players. When you grind the penny slots, your average payback rate stands right around 90% — giving the house a huge edge of 10%.

But when you bump the stakes up only slightly to a nickel per spin, the payback rate climbs to over 94% on average. That figure inches even higher on the $1 machines, so whenever possible, ditch the penny slots for a much better play.

4 – Don’t Play Drunk, High, or Tired to Avoid Making Mistakes

This one sounds simple enough, but when the complimentary cocktails start flowing, even a normally reasonable gambler can start betting with reckless abandon.

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The casinos count on that too, so don’t cooperate with their attempts to further separate slot players from their money. Staying sober and alert at the machines is the best way to prevent poor decisions, and poor decisions by players represent the casino industry’s lifeblood.

5 – Take a Visual Tour of How Slot Machines Really Work

You can’t slay a dragon without studying its tendencies first, and that holds especially true for slot enthusiasts.

Unless you know exactly how the modern video slots and RNGs really work, it can be far too easy to succumb to superstition.

That’s why I recommend using this extremely cool 3-D, 360-degree visual tour of a slot machine’s inner workings.

6 – Study a Slot Machine Glossary to Learn the Latest Lingo

If you’re still wondering about the difference between paylines and payback percentage, or progressive vs. fixed jackpots, we can help.

Consider this detailed slot machine glossary to be required reading.

Slot machines are fun, easy to learn, and wildly entertaining when the reels are cooperating. On the other hand, most slot games are designed to put players in an almost inescapable hole, with payback percentages hovering around 90% to create the ultimate longshot.

Knowing how to avoid the worst machines while bringing your money to the most player-friendly slots is half the battle, so keep these six tips in mind the next time you embark on that jackpot journey.

One final tip is for those of you how play slot machines online. Make sure that you stick to safe and trusted online casinos, as you’ll get a much better experience overall.

Appendices
Slots Analysis
Miscellaneous

Introduction

Value Of Old Slot Machines

Click on this image to play.

I designed the Atkins Diet slot machine in 2008 for a speaking engagement at a conference on slot machines in Sydney, Australia. It was used as an example of how video slots, sometimes known as pokies, are designed. I'll start with presenting the pay table and reel strips, and then I'll show the probability and return of the various ways to win.

Pay Table

The next table shows the Atkins Diet pay table. All wins must be left aligned. The Atkins symbol is wild and can substitute for any symbol except the scale.

Atkins Diet Pay Table

Symbol	5 in a row	4 in a row	3 in a row	2 in a row
Atkins	5000	500	50	5
Steak	1000	200	40	3
Ham	500	150	30	2
Buffalo Wings	300	100	25	2
Sausage	200	75	20	0
Eggs	200	75	20	0
Butter	100	50	15	0
Cheese	100	50	15	0
Bacon	50	25	10	0
Mayonnaise	50	25	10	0

Reel Strips

The following are the reel strips for Atkins Diet.

Atkins Diet Reel Strips

Stop	Reel 1	Reel 2	Reel 3	Reel 4	Reel 5
1	Scale	Mayonnaise	Ham	Ham	Bacon
2	Mayonnaise	Buffalo Wings	Butter	Cheese	Scale
3	Ham	Steak	Eggs	Atkins	Steak
4	Sausage	Sausage	Scale	Scale	Ham
5	Bacon	Cheese	Cheese	Butter	Cheese
6	Eggs	Mayonnaise	Mayonnaise	Bacon	Sausage
7	Cheese	Ham	Butter	Cheese	Butter
8	Mayonnaise	Butter	Ham	Sausage	Bacon
9	Sausage	Bacon	Sausage	Steak	Buffalo Wings
10	Butter	Steak	Bacon	Eggs	Cheese
11	Buffalo Wings	Sausage	Steak	Bacon	Sausage
12	Bacon	Mayonnaise	Buffalo Wings	Mayonnaise	Ham
13	Eggs	Ham	Butter	Sausage	Butter
14	Mayonnaise	Atkins	Mayonnaise	Cheese	Steak
15	Steak	Butter	Cheese	Butter	Mayonnaise
16	Buffalo Wings	Eggs	Sausage	Ham	Eggs
17	Butter	Cheese	Eggs	Mayonnaise	Sausage
18	Cheese	Bacon	Bacon	Bacon	Ham
19	Eggs	Sausage	Mayonnaise	Buffalo Wings	Atkins
20	Atkins	Buffalo Wings	Buffalo Wings	Sausage	Butter
21	Bacon	Scale	Ham	Cheese	Buffalo Wings
22	Mayonnaise	Mayonnaise	Sausage	Eggs	Mayonnaise
23	Ham	Butter	Bacon	Butter	Eggs
24	Cheese	Cheese	Cheese	Buffalo Wings	Ham
25	Eggs	Bacon	Eggs	Bacon	Bacon
26	Scale	Eggs	Atkins	Mayonnaise	Butter
27	Butter	Buffalo Wings	Buffalo Wings	Eggs	Steak
28	Bacon	Mayonnaise	Bacon	Ham	Mayonnaise
29	Sausage	Steak	Butter	Sausage	Sausage
30	Buffalo Wings	Ham	Cheese	Steak	Eggs
31	Steak	Cheese	Mayonnaise	Mayonnaise	Cheese
32	Butter	Bacon	Steak	Bacon	Buffalo Wings

Example Spin

Let's look at the following example spin.

The way multi-line video slots work is to pick one random number for each reel and map it onto the corresponding stop on the reel strip. Let's assume that the random number chosen designates where the reel will stop on the middle row. The symbols above and below will be those directly above and below on the reel strips.

Random numbers of 27, 14, 3, 31 and 27, for reels 1 to 5 respectively, would have resulted in the example above. The following table shows the position on the reel strips where all 15 visible symbols fall on the screen.

Atkins Diet Reel Strips

Stop	Reel 1	Reel 2	Reel 3	Reel 4	Reel 5
Top	26	13	2	30	26
Middle	27	14	3	31	27
Bottom	28	15	4	32	28

It should also be noted that the reel strips wrap around. So, if the number chosen for the middle position is 1, then the top position would be 32. Likewise, if the middle position is 32, then the bottom position would be 1.

Symbol Distribution

The first step to calculating the return of a video slot is to count how often each symbol appears on each reel. The following table answers that question.

Atkins Diet Symbol Distribution

Symbol	Reel 1	Reel 2	Reel 3	Reel 4	Reel 5
Atkins	1	1	1	1	1
Steak	2	3	2	2	3
Ham	2	3	3	3	4
Buffalo Wings	3	3	3	2	3
Sausage	3	3	3	4	4
Eggs	4	2	3	3	3
Butter	4	3	4	3	4
Cheese	3	4	4	4	3
Bacon	4	4	4	5	3
Mayonnaise	4	5	4	4	3
Scale	2	1	1	1	1
Total	32	32	32	32	32

Line Pay Math

There are 32 × 32 × 32 × 32 × 32 = 33,554,432 possible outcomes in Atkins Diet. The following table shows how many of these combinations result in each possible win. These figures are the result of a program with five nested loops that tallied the total for each win for each possible combination.

Atkins Diet Line Pay Combinations

Symbol	5 in a row	4 in a row	3 in a row	2 in a row	Total
Atkins	1	28	513	1,024	1,566
Steak	431	2,996	32,480	326,656	362,563
Ham	955	5,157	42,112	315,392	363,616
Buffalo Wings	764	5,348	58,464	430,080	494,656
Sausage	1,595	8,613	54,432	-	64,640
Eggs	956	6,692	52,864	-	60,512
Butter	1,995	10,692	88,704	-	101,391
Cheese	1,996	13,860	85,536	-	101,392
Bacon	2,976	20,832	103,168	-	126,976
Mayonnaise	2,980	20,860	128,736	-	152,576
Total	14,649	95,078	647,009	1,073,152	1,829,888

The next table shows the probability of each possible win. This is the number of winning combinations for that win, shown in the table above, divided by the total number of combinations of 33,554,432. The total in the lower right cell reflects a hit frequency per line (probability of winning anything) of 5.45%.

Atkins Line Pay Probabilities

Symbol	5 in a row	4 in a row	3 in a row	2 in a row	Total
Atkins	0.00000003	0.00000083	0.00001529	0.00003052	0.00004667
Steak	0.00001284	0.00008929	0.00096798	0.00973511	0.01080522
Ham	0.00002846	0.00015369	0.00125504	0.00939941	0.01083660
Buffalo Wings	0.00002277	0.00015938	0.00174236	0.01281738	0.01474190
Sausage	0.00004753	0.00025669	0.00162220	-	0.00192642
Eggs	0.00002849	0.00019944	0.00157547	-	0.00180340
Butter	0.00005946	0.00031865	0.00264359	-	0.00302169
Cheese	0.00005949	0.00041306	0.00254917	-	0.00302172
Bacon	0.00008869	0.00062084	0.00307465	-	0.00378418
Mayonnaise	0.00008881	0.00062168	0.00383663	-	0.00454712
Total	0.00043657	0.00283355	0.01928237	0.03198242	0.05453491

The next table shows the contribution to the return for each possible win. Each cell is the product of the probability, from the table above, and the pay table. The right cell shows a total return from line pays of 63.46%.

Atkins Diet Expected Returns

Symbol	5 in a row	4 in a row	3 in a row	2 in a row	Total
Atkins (wild)	0.000149	0.000417	0.000764	0.000153	0.001483
Steak	0.012845	0.017858	0.038719	0.029205	0.098627
Ham	0.014231	0.023054	0.037651	0.018799	0.093734
Buffalo Wings	0.006831	0.015938	0.043559	0.025635	0.091963
Sausage	0.009507	0.019252	0.032444	-	0.061202
Eggs	0.005698	0.014958	0.031509	-	0.052165
Butter	0.005946	0.015932	0.039654	-	0.061532
Cheese	0.005949	0.020653	0.038238	-	0.064839
Bacon	0.004435	0.015521	0.030746	-	0.050702
Mayonnaise	0.004441	0.015542	0.038366	-	0.058349
Total	0.070029	0.159124	0.331651	0.073792	0.634597

Scatter Pay

The scale serves as a scatter pay symbol. The player gets credit for a scatter pay if it appears anywhere on the screen. Unlike line pays, scatters don't need to be left aligned or on an active pay line. To be fair, scatter pay wins are based on the total amount bet. The following table shows the number of scatters per reel, total symbols per reel, and the probability of a scatter on that reel, which is 3×(column 2)/(column 3). The reason for multiplying by 3 is there are three visible positions on each reel.

Atkins Diet Scatters per Reel

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Reel	Scatters	Total Symbols on Reel	Scatter Probability
1	2	32	0.18750
2	1	32	0.09375
3	1	32	0.09375
4	1	32	0.09375
5	1	32	0.09375

The next table shows the probability and contribution to the return for 0 to 5 total scatter symbols. The lower right cell shows that the scatter pay feature returns 6.98% to the return of the game.

Atkins Diet Scatter Pay Probabilities and Returns

Scatters	Pays	Combinations	Probability	Return
5	100	486	0.000014	0.001448
4	25	20,898	0.000623	0.015570
3	5	353,916	0.010548	0.052738
Total	33,554,432	0.011185	0.069756

Bonus

The player earns the bonus if he gets at least three scattered scales. The sum of the probability of 3 to 5 scales in the table above is 0.011185. During the bonus, the player will get ten free spins, and all wins are tripled.

In this game, free spins use the same reel strips as the base game. We have already shown that the expected line pay return is 0.634597 and the expected scatter return is 0.069756. So, if the win is tripled, then the expected win per spin is 3 × (0.634597 + 0.069756) = 2.113058.

Next, there is a rule that free spins can be won in free spins, infinitely. The expected number of total spins in a bonus is 10/(1-10×0.011185) = 11.259335.

So, with 11.259335 free spins, and an average win per spin of 2.113058, the expected final win per bonus is 11.259335 × 2.113058 = 23.791632.

The probability of winning the bonus in the first place is 0.011185, so the expected return per initial spin of the bonus feature is 0.011185 × 23.791632 = 0.266105.

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Summary

The following table summarizes the three ways to win in the game. The bottom right cell shows an expected return of 97.046%. In other words, for every dollar the player bets, he can expect to get back 97¢.