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Slots - General Questions

If a slot machine had five reels, and the probability of getting a cherry were the same on each reel, what would be the probability of getting any specified number of cherries on a spin?

Jay

Let's let p be the probability of getting a cherry on any given reel and n be the number of cherries on the payline. The probability of getting n cherries is combin(5,n) * pn * (1-p)5-n. Combin(5,n) denotes the number of ways that n cherries can appear on five different reels. In particular, combin(5,0)=1, combin(5,1)=5, combin(5,2)=10, combin(5,3)=10, combin(5,4)=5, and combin(5,5)=1. This function can be used directly in Excel and is explained in more detail in my section on probabilities in {poker}. However, to take a specific example, if the probability were 5% of getting a cherry on any given reel then probability of getting 3 cherries would be 10 * .053 * .952 = 0.001128125 .
How is it possible for casinos to program their slot machines to pay out a regulated percentage if the slot machines run off a random number generator? It would seem there is no way to know what number or combination (winning or losing) is next.

Steve from Milton, U.S.

The casinos don't actually program the casinos to pay a certain percentage, but determine the weighting of the reels so that the theoretical return is whatever they wish. In the short run, the actual return can be either much higher or lower than the theoretical return. However, the laws of mathematics dictate that the actual return will get closer to the theoretical return the greater the number of trials.
What is the best method for receiving comps from slot attendants when playing slot machines?

Donald from Golconda, USA

This is not my area of expertise. However Jean Scott "the Queen of Comps" says you should establish a relationship with a casino host anywhere you plan to play a lot. Then ask them for a comp after you have given them sufficient play.
Did old slot machines, that were not computer operated have greater chances of winning? If so, how did they work?

Sarah from Oshawa, Ontario, Canada

I don’t know whether the chances of winning were better or not. They worked the same way as they do now except each stop on each reel had an equal chance. The very early ones didn’t pay money but chewing gum, which explains the bar symbols (sticks of gum) and fruits (flavors) on some modern slot machines.
How does the RNG logic interact with the "hold %"? I believe that most machine are set to hold X% and that over time that number will be reached. It would seem that that "hold" number must have some influence on the odds a machine will pay or not. I realize that a newly installed machine that has never been played can hit the royal on the first play, however, it is my belief that over time, the hold % will met by that machine. I have also heard the term "cycle". I slot tech at a casino told me that a machine was on a 365-day cycle. What does that mean?

anonymous

First lets clear up what the term "hold" means. For purposes of electronic games it is the theoretical return the game is set to. In both video poker and slots each play is random and independent of all past plays. The laws of mathematics dictate that even with independent trials the as the sample size gets larger the actual return will tend to get closer to the theoretical mean, or the hold. So contrary to popular belief a machine never goes hot or cold to get back in balance. Never mind the term "cycle." It is a poorly named industry term for the number of possible outcomes of the random number generators inside the machine. Unfortunately the term has trickled down only to confuse low level employees and players alike. Contrary to popular myth there are no cycles and again each play is equally random and independent of all other plays.
In which order would you place these machines for playing or not playing? Thanks. Double Diamonds, Triple Diamonds, Triple Wild Cherry, Wild Cherry.

anonymous

I would rank them all equally. Generic 3-reel slots tend to be set to about the same return for the same casino and coinage.
On slot machines with bonus rounds like "Jackpot Party" or the one with the leprechauns are the winning boxes or pots of gold predetermined the second you hit the bonus round or on an individual basis by a random number generator when you select each box.

Jerry from Tacoma

It could be done either way. I think most slot makers prefer to have the prizes constantly mixed up, so the outcome is a function of both which box you pick and when you pick it. Mathematically speaking it doesn’t make any difference which way of the two they program it.
After recently visiting Las Vegas for the first time in 15 years, I noticed that the slot machines at the casinos I visited accepted only credit slips, cards or dollar bills, not coins. Have all of the slots become this way?

Margueritte from Los Angeles

Most of them are now “ticket in ticket out” as they say, or just TITO. Personally, I applaud this change. No more dirty hands and waiting for hopper fills. Some of the older casinos downtown still use coins and tokens, but I doubt for much longer.
How often do Las Vegas or Reno (Nevada as a whole if easier) change their slot machines? Better stated perhaps as what is the average life cycle of a slot machine before it is worn out, out of fashion, unsecure, etc.? How does that average compare to smaller markets such as Deadwood, South Dakota?

Justin from Rapid City

I forwarded your question to Brian, a former regulator and current casino manager. Here is what he said.

There are two types of changes. The first would involve completely swapping out the machine and the second would consist of simply changing the game, but keeping the existing cabinet. As you can imagine, changing the software is much cheaper which is why there is so much hype around downloadable games. How often games are swapped out depends on a casinos capital expenditures budget. Participation machines are turned over much more rapidly because the manufacturer has a vested interest in keeping the best product on the floor. In many instances, they will handle the scheduling for software and new machine replacements. Participation machines are those that are on lease to the property by the manufacturer. Usually, the manufacturer gets 20% of the revenue, less taxes. From an accounting perspective, the useful life of a slot machine is 5 years and then the asset is fully depreciated (no longer has a book value). The final consideration is popularity. How often do you go into a casino and see a section of slot machines that are the old IGT three reel Red White and Blue machines? If the machines are performing well, why spend $10,000+ to replace each unit?

About the February 1, 2006 column

anonymous

Follow up: In the February 1, 2006 column, a reader was mad because of the short expiration dates on slot machine tickets. I took her side, saying they shouldn’t expire at all. Many readers took me to task, saying that casinos routinly honor expired tickets. So I then did an experiment in which I collected $2 tickets up and down the Strip. After they expired I went to cash them in all of them were honored. So I have amended my answer and I offer my apologies to the casinos for my earlier harsh words.
In the Realtime Gaming slot machine Fruit Frenzy you always get the maximum 25 paylines activated in the free spin bonus, regardless of how much you bet on the initiating spin.

You're always saying there is no skill in slots, but wouldn't the player get a better return betting one line only in this game, since the value of the bonus is the same regardless of how many coins the player bet?

Oren from Thailand

You're right that the value of the bonus is the same regardless of how many lines the player bets in the initiating spin. However, it takes five bonus symbols on an ACTIVE payline to trigger the bonus. So, the probability of triggering the bonus is directly proportional to the number of lines bet. Thus, the return to the player should be the same regardless of the number of lines bet.
The Hot Roll bonus round on slot machines awards the player the following number of coins according to the total of two dice. The player keeps collecting until he rolls a total of seven, which ends the bonus. If he rolls a seven on the first roll, he gets a consolation prize of 70 coins. Following are the prizes for all other totals besides seven:

  • 2 or 12: 1,000
  • 3 or 11: 600
  • 4 or 10: 400
  • 5 or 9: 300
  • 6 or 8: 200


My question is what is average bonus win?

anonymous

Click the following button for the answer.

The answer is 1983.33.

Click the following button for the solution.

Let x be the answer. As long as the player doesn't roll a seven he can always expect future wins to be x, in addition to all previous wins. In other words, there is a memory-less property to throwing dice in that no matter how many rolls you have already thrown you are no closer to a seven than you were when you started.

I won't go into the basics of dice probabilities but just say the probability of each total is as follows:
  • 2: 1/36
  • 3: 2/36
  • 4: 3/36
  • 5: 4/36
  • 6: 5/36
  • 7: 6/36
  • 8: 5/36
  • 9: 4/36
  • 10: 3/36
  • 11: 2/36
  • 12: 1/36


Before considering the consolation prize, the value of x can be expressed as:

x = (1/36)*(1000 + x) + (2/36)*(600 + x) + (3/36)*(400 + x) + (4/36)*(300 + x) + (5/36)*(200 + x) + (5/36)*(200 + x) + (4/36)*(300 + x) + (3/36)*(400 + x) + (2/36)*(600 + x) + (1/36)*(1000 + x)

Next, multiply both sides by 36:

36x = (1000 + x) + 2*(600 + x) + 3*(400 + x) + 4*(300 + x) + 5*(200 + x) + 5*(200 + x) + 4*(300 + x) + 3*(400 + x) + 2*(600 + x) + (1000 + x)

36x = 11,200 + 30x

6x = 11,200

x = 11,200/6 = 1866.67.

Next, the value of the consolation prize is 700*(6/36) = 116.67.

Thus, the average win of the bonus is 1866.67 + 116.67 = 1983.33.

How can I calculate the number of combinations of each win for slots with "Multiway wins"? You can assume that I have the reel strips.

James from Vermont

For the benefit of other readers, slots with "Multiway" wins cover all possible pay-lines. However, the game will pay only once for each way combination through winning symbols. Once a reel is reached with no winning symbols, the pay-lines end there.

Let's look at an example based on a game with five reels and three visible rows. All wins are left aligned only. Suppose the player had a winning symbol on reels 1, 2, 3, and 5. The player would be paid only once for three of that symbol. It does not matter that there are 9 ways to run through reels 4 and 5, because in this example the pay-lines end with reel 3.

Next suppose the player had the same winning symbol this many times on each reel:

  • Reel 1: 2
  • Reel 2: 1
  • Reel 3: 3
  • Reel 4: 2
  • Reel 5: 1


The player would be paid for 2×1×3×2×1 = 12 pay-lines.

If the player covered the entire screen with the same winning symbol, he would be paid for 35=243 pay-lines.

Next, let's move onto the answer. Let's assume there are wins for 3 to 5 symbols only.

Let's define some terminology:

  • tx = total reel stops on reel x.
  • nx = total count of the winning symbol on reel x.
  • px = positions on reel strips x where there is no winning symbol visible on the reel.


    • For reel 3 the answer is 33 × n1 × n2 × n3 × p4 × t5.

    For reel 4 the answer is 34 × n1 × n2 × n3 × n4 × p5.

    For reel 5 the answer is 35 × n1 × n2 × n3 × n4 × n5.

I'm sure you've heard of the guy who claims to know of a bug in Aristocrat slot machines and is offering to tell them about it, for a price. If you knew of such a bug, how would you go about getting the most money out of it?

  • A) Hit a big casino hard with a few max bets, then leave for a few weeks before hitting another big casino. Keep going until they fix the hole or back you off.
  • B) Intersperse medium sized winning bets with small losers all over town and milk the cows for hopefully a longer period of time.
  • C) Form a team and do (A).
  • D) Form a team and do (B).
  • E) Contact the manufacturer and offer it to them for a finders fee + a residual.
  • F.) Other?

Ayecarumba

Yes, I've heard that story. For the benefit of my other readers, here is a Wired link to the story: Meet Alex, the Russian Casino Hacker Who Makes Millions Targeting Slot Machines.

Putting moral issues aside, and assuming getting caught is not a major concern, I would go with choice B. I would have a hard time trusting a team to report winnings honestly and not reveal the secret. Seems to me flying under the radar would be the best choice.

Do you have any programming tips for doing the math on a "Multi-way" slot machines like Buffalo?

Mike H. from New Jersey

First, let me explain the question for the benefit of other readers. A popular new way for slot machines to pay is according to every "way" on the screen. A "way" is every unique set of positions that go through a paying combination, counting only reels that are part of the win. Let's look at the following picture as an example from a Buffalo slot:

This game as 45=1,024 "ways" to win, because there are four positions shown on each reel. However, in this case, only two ways pay, for two buffalo each. Both include the one buffalo on reel 1. Then 1 way for the buffalo on row 1 of reel 2 and a second way for the buffalo on row 2 or reel 2. Although there are 43 = 64 ways the game could go through positions on reels 3 to 5, it doesn't matter. "Ways" only count the ways to go through symbols that contribute to the win.

With that explanation out of the way, let me introduce some functions for my answer to your question:

  • Let v = number of visible rows on the machine.
  • Let n(s,r) = number of times symbol s or a wild appears on reel r.
  • Let b(s,r) = number of sequences on reel r with no appearances of symbol s or a wild symbol that may substitute for s. In other words, sequences blocking any wins for a given symbol on the reel because it doesn't appear.
  • Let t(r) = Total length of reel r


That said, here are the number of winning combinations according to the number of winning symbols in the win:

  • The number of combinations for a five-symbol win of symbol s equals n(s,1)*n(s,2)*n(s,3)*n(s,4)*n(s,5)*v^5.
  • The number of combinations for a four-symbol win of symbol s equals n(s,1)*n(s,2)*n(s,3)*n(s,4)*b(s,5)*v^4
  • The number of combinations for a three-symbol win of symbol s equals n(s,1)*n(s,2)*n(s,3)*b(s,4)*t(5)*v^3
  • The number of combinations for a two-symbol win of symbol s equals n(s,1)*n(s,2)*b(s,3)*t(4)*t(5)*v^2
How much would an extra guaranteed wild be worth playing a video slot machine?

anonymous

That depends on lots of things. The way I will interpret your question is what is the value of an extra wild above the average number the player would normally get. While the answer will vary significantly from game to game, a significant factor is the number of rows on the screen. If there are three rows, the extra wild will benefit 1/3 of the paylines. Likewise, if there are four rows, it will be less valuable, affecting 1/4 of the paylines.

To answer your question, I looked at the game Cleopatra, which I already deconstructed. The following table shows the increase in the expected value for a wild, compared to a random number of wilds.

Value of Extra Wild in Cleopatra

Reel 3 Rows 4 Rows
1 95.71% 71.79%
2 99.76% 74.82%
3 76.24% 57.18%
4 21.25% 15.93%
5 1.96% 1.47%

Thank you for your analysis of must-hit-by progressives. My question is does your formula for the hit point to play assume an immediate player advantage or a situation that may be slightly negative at first, but will shortly turn positive as the player contributes to the meter?

anonymous

Good question. It previously gave a formula for a "short term" player, where the jackpot must be positive on bet one.

However, for the long-term player, who can afford to play until the jackpot hits, the hit point is less. I have updated the page to include formulas for both types of players. Briefly, the two formulas are:

j (short term) = m × (1-f)/(1-f+r)
j (long term) = m × (1-f-r)/(1-f+r)

Where:

j = Breakeven jackpot size (with 0% house edge)
f = Value of all fixed wins plus slot club points and incentives.
m = Maximum jackpot (the must-hit-by point)
n = Minimum jackpot (the reseed point)
r = Rate of meter rise

I hear the minimum taxable slot machine jackpot has been increased from $1,200 to $5,000. Is this true?

Heather

No.

A bill has been introduced to do just that. However, most bills go nowhere and never even get voted on, so slot players should not get their hopes up.

Personally, if I had a vote in the matter I would vote YES enthusiastically. The current $1,200 limit has not been increased since 1977. According to the American Institute or Economic Research, $1,200 in 1977 would have been worth $5,336 in 2021. I could easily make an argument against all taxation of gambling winnings. However, I know that ship won't float, so the least we can do is recalibrate the minimum taxable jackpot to inflation, which should have been done all along.

For more information, please read Bill would raise slot jackpot tax report threshold to $5,000 at the Las Vegas Sun.