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Slot Machine Probability: a Mathematical Approach to Slot Machines
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Today, the mathematics of slot machines.
The University of Houston mathematics department presents this program about the machines that make our civilization run, and the people whose ingenuity created them.
Mathematicians first got interested in randomness by studying games of chance.
Ever since, the histories of mathematics and gambling have been intertwined.
Clever gamblers use mathematics to look for the smallest advantages, and casinos use sophisticated mathematical tools to devise new ways of drawing in players.
Indeed, a patent granted to the Norwegian mathematician Inge Telnaes in 1984 transformed the gambling industry.
The likelihood of getting three cherries, and winning the jackpot, is 1 in 1,728.
However, the only way to increase the payout slot machine math to decrease the chances of hitting a jackpot.
Adding another reel is a possibility.
But people do not like machines with more reels � they intuitively, and rightfully, feel that extra reels diminish their chance of winning.
Another possibility is to put more symbols on each reel.
But the astronomical jackpots you see in casinos today would then require truly enormous machines.
Inge Telnaes proposed a simple solution: Let a random number generator � a computer chip � determine the combination of symbols that appear when the reels stop.
In other words, use a chip to control where the reels stop on a spin, but create the illusion that the wheels stopped on their own.
The number of possible outcomes on the slot machine does not change.
However, by reprogramming the chip, the visit web page has full control over the likelihood of each of the different outcomes.
For instance, the operator could make the three cherries appear only once in a million spins.
This was a brilliant insight: Suppose I pick a number between one and a million.
Would you be willing to bet that you can guess that number?
The answer is probably not.
But let a computer chip pick such a number, put the chip in a machine with blinking lights and spinning reels, and many people will be more than willing to make the bet.
It is simply because what people assume is happening in a slot machine consider, free slot game machine seems very different from what is actually happening.
The Magician oil painting by Hieronymus Boschfrom between 1475 and 1480 The history of gambling is also intertwined with that of a less reputable group � tricksters and swindlers.
In the long run, the only sure way to make money by slot machine math is to create slot machine math illusion that your opponent https://pink-stuf.com/slot-machine/train-slot-machine.html win, while keeping the odds firmly on your side.
And that gives those who know math a very solid advantage.
I'm Kresimir Josic, at the University of Houston, where we're interested in the way inventive minds work.
Theme music NOTE: In slot machine math example with three cherries, I assumed that one only wins in the case the spin results in three cherries, and there is no other winning combination.
In actuality, there are typically many winning combination, and as a result, the jackpot would have to be even smaller.
The following story in Wired Magazine shows the drawbacks of the new generation of slot machines � they are easier to hack and to counterfit than their mechanical counterpart.
Here is a more exhaustive discussion of the history of slot machines, and the random number generators within them.
You may slot machine math to scroll towards the end of the article to read about how flaws in the design of gambling machine resulted in somebody picking 19 out of 20 winning numbers in a game of KENO � and doing so 3 times in a row.
Both images are from Wikipedia.
The slot machine image was taken by Jeff Kubina.
For more mathematics in everyday life, visit.
This episode was first aired on September 7th, 2011.

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The mathematics of slots
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The mathematics of slots
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???????????????????? ????????????????????????????? ????? 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.
Although I have never played on one knowinglyyou could have a slot machine were each slot itself has a different amount of symbols.
Calculating the probability of winning on a slot machine is fairly simple.
The harder the combination, the bigger the reward.
In gambling, the term payout percent is used when describing how "lucrative" the slot machine is.
The payoff acts as a weight for each winning combination.
It is like weighted average, but not quite.
Let us look at analyzing a pair of slot machines.
For this blog entry, we will assume that the slot machine has one line and takes only one coin per roll.
Analyzing a Simple Slot Machine The first machine has three slots.
On each slot there is an apple, orange, lemon, banana, melon, and a Joker.
II : Three of a kind that isn't a Joker.
One the first slot, you can get any of the five fruit apple, orange, lemon, banana, melon.
Once that fruit is selected, that fruit has to match on the next two slots.
So, there are 5 such combinations: three apples, three oranges, three lemons, three bananas, and three lemons.
III : Jokers can appear any two slots: slots 1 and 2, slots 1 and 3, and click here 2 and 3.
The other slot must contain fruit.
IV : Jokers can appear in any slot, the other two slots must have fruit.
In this instance, it does not matter if the other two slots match.
The table below shows the calculation of slot's payout.
That is pretty good in terms of slot machines.
It could be a little misleading given that the harder combinations are given more weight.
I doubt no one would play a slot machine with less than a 50% slot machine math of spinning a winning if there was no added incentive, such as a richer payoff.
A More Complex Slot Machine We are still at one line, three slots.
Here is the slot distribution: There are 23 symbols per slot.
Also note that the non-equal distribution of symbols on each slot.
With the lack of BARs, cherries, and Sevens 7it is obvious that getting winning combinations involving these three symbols offer higher playoffs.
The winning combinations, with payoffs are: See the table above for the distribution of symbols I 3 BARs.
Number of winning combinations: 1 II 3 Sevens.
If everything paid 1 coin equally, it would drop to just over 42%.
For more complex slot machines more slots and multiple linesthe use of a spreadsheet or other computer program will come to the aid of calculating payoff since the number of calculations grow with complexity.
Eddie Source: Slot Machine Math - Gambler's Bookcase This blog is property of Edward Shore.
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Nice articleBut i do have one query after studying you More complex slot tableas you have mentioned the possible outcome of every symbol regarding every reel so do i need to make use of different set of arrays for every slot as you described above the possibility of occurrence of Cheery symbol in slot one is 3 and in slot two and three is 1 so this means i have to insert two different arrays dynamically in reel while clicking spin buttonevery slot works with different array items which consist of different number of cheery symbols in it?
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Stay tuned with the most reliable Slots USA Online at SlotPlus US It is good to understand probability on the Slot machines.
It is key to remember that they are based on Random Number Access tables.
Also each Reel may have free slot game machine e.
Each space or symbol can have quite a few random Numbers associated with it.
If there are 100 random numbers per reel on a 3 reel game, you now have 1,000,000 possible combinations.
As soon as you play the random combination is set before the reels actually revolve.
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Regarding calculating the odds, what if you have 1 wild card on each reel that can replace any symbol.
How do you calculate the odds in that case?
You just put 1 extra of each symbol in the equation?
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The opportunity to play a casino game is a product that the consumer purchases with his gaming dollar.
The view I take slot machine math that the player wants to have a certain experience when he sits at a game and is willing to pay for that experience by means of the inevitable bite the house edge takes.
To slot machine math understand the product the player is purchasing requires a journey into the mathematics of.
Marketing must understand this material when developing effective strategies.
X-FOR-1, X-TO-1, 1-IN-X Most table game wagers are paid in an "X-to-1" fashion.
What this means is that if the player wins the round, his wager is returned and he is paid an additional amount for his win.
If the player loses, then his wager is removed from the table.
For example, if the player bets on red-7 straight-up in then the payout is 35-to-1.
This type of pay is different from slot machines.
Forevery time the player pushes the button slot machine math pulls the handle, an amount is deducted from his credit balance.
Then the reels spin, and when they stop he is paid an amount based on the pattern displayed by the reels by slot machine math to some pay table.
We say that slots are "X-for-1.
Blackjack slots typically show a payout of 2 credits when the player is dealt a blackjack.
The player knows that blackjack usually pays 3-to-2, so this seems like a sweet deal.
However, this payout is 2-for-1.
Thus the player forfeits his original 1 credit and then wins back 2 credits if he is dealt a blackjack.
The player nets 1 credit in this way.
Thus 2-for-1 is the same as 1-to-1.
Few would play the table game of blackjack if the payout for a blackjack was reduced to 1-to-1, yet people often play these slots with a payout of 2-for-1.
The progressive wager in Caribbean Stud is an example of an "X-for-1" wager appearing in a well known.
The player's wager is automatically collected as it is made, and a light illuminates on the table indicating the player has made this wager.
Another common "X-for-1" wager is the bad-beat jackpot in the poker room.
Return please click for source player's original wager if the player wins or pushes; otherwise collect the player's wager.
For example the Pair Plus wager in has a straight pay table based on the three cards dealt to the player.
For pc slot machine games software, in craps, the shooter will get a 7 over the long run 1-in-6 rolls.
In double-0 Roulette, the number red-7 will come up over the long run 1-in-38 spins.
The player is dealt a blackjack over the long run approximately 1-in-21 hands.
It is used to describe the rate of occurrence of an event at a table game.
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Russell Hunter Publishing Inc Slot Machine Math All gaming machines are designed to pay the player back a percentage of what is played.
The amounts vary from machine to machine and from casino to free igt slot games />All machines have one thing in common: The longer the machine is played, the closer the actual payouts will be to the theoretical results.
Slot machines use a random selection process to achieve a set of theoretical odds.
Random selection means that each time the lever is pulled and the reels are set in motion a combination of symbols are randomly selected.
The "random" aspect ensures that each pull of the handle is independent from every slot machine poker pull, so that the results of the previous pull, and the one before that, have no effect on the current one.
The theoretical odds are built into the design and program of the machine, and it is possible to calculate the exact payout percentage for any machine over the long-term.
Except for the video slots, slot machines have wheels called reels with slot machine math printed on each wheel.
Each reel symbol represents a stop which may come to rest on the payline, and may or may not be part of a combination of symbols resulting in a payoff.
The likelihood of winning any payoff on any slot machine is related slot machine math the number of reels and the number of symbols on each reel.
The most common type of mechanical slot machine has three reels with twenty symbols on each reel.
To calculate the total number of combinations of symbols on this machine, we multiple the number of stops symbols on each reel by the number of stops on each of the remaining reels.
Likewise, we can calculate the probability of any combination of symbols hitting if we know the number of times each symbol appears on each reel.
When mechanical slots dominated, it was not too difficult to count the symbols on each reel and determine exactly the payoff of a given machine.
With microprocessor controlled slots this task has become almost impossible, as the number of stops per reel can be as many https://pink-stuf.com/slot-machine/star-joker-slot-machine.html 256.
To determine the payoffs of such a machine would require significant reverse engineering and is beyond the scope of almost every player.
The number of reels has slot machine math greater effect on the probabilities than the number of symbols per reel.
Every slot machine has a predetermined payout percentage.
When you hear things like "our slots pay back 98.
Conversely, we could state that as for every dollar played, the casino will retain 1.
These percentages only hold true over very long-term play consisting of hundreds of thousands or even millions of plays.
There are a couple of things wrong with this line of thinking.
First, theoretical percentages will be attained only over long periods of play.
visit web page a few dozen, or even a few hundred rolls, the payback percentage will vary greatly.
Most people will buy twenty dollars worth of tokens and continue to play with this money until it is gone.
After inserting the first round of coins in the machine, they will continue playing with any coins left in the tray, and they will continue this pattern until no coins are left.
And then they wonder slot machine math it was possible for a 98.
The answer is that the casino continues to extract its percentage on every coin inserted slot machine math the machine.
The player will not limit his play to twenty dollars or one hundred dollars but will continue to redeposit coins.
The machine will, at least over the long-term, continue to grind away at all money played.
Table 6 shows the devastating effect the house edge can have on the player's bankroll.
We can see the power of a hold rate super slot gratis poker 15%.
Even though we have gradually lost some money slot machine math this machine, we can see that hitting a single higher payoff would put us ahead and that we have gained much more playing time to do so.
If you never cared what the casino hold was before, this should open your eyes.
It is imperative that you always seek and play machines with lower hold percentages.
Unfortunately, casinos do not label their machines with the hold or payback percentages.
However, the player's win rates are available for different locations.
Table 7, which shows the win rates for different United States casino locations, was derived from information published by the Casino and Gaming Control Boards and Commissions of Nevada, New Jersey, Illinois, Iowa, Connecticut and Colorado.
If we were seeking to play the highest payback slots, we might begin our search with this information.
The highest average payback in the U.
The average for all Nevada casinos is 95.
The next best place to play is in Colorado, with an average rate of 93.
Moving east, we find that the Illinois river boats offer the third best choice at 92.
We also notice that the win rates increase with the size of coin accepted, with the nickel slots paying out the lowest percentages and the five dollar slots the highest.
In More info, the win rates average about click here for all but the lowly nickel slots.
So, on the average, it is safe to say that if we limit our play to Nevada casinos and avoid the nickel slots, we can expect to receive about a 95% payback.
Conversely, if we limit our play to nickel slots on the Illinois river boats, we will average a measly 89.
If we are a quarter player, then Downtown Las Vegas is the best deal in the U.
We will analyze a 3 reel, two coin multiplier which pays bonuses on two of its payoffs.
The pay schedule for this machine is shown in Table 8.
The first step in analyzing this machine is to break out the number of symbols stops per reel.
This particular machine has 32 stops per reel and the reel analysis is shown in Table 9 Table 9.
Since bonuses are offered when the second coin is played, we will add another 32,768 different combinations with play of the second coin.
On this machine, we will use totals of 32,768 combinations with one coin played and 65,536 combinations when two coins are played.
Table 10 shows an analysis of all winning combinations on this machine.
Analysis of Winning Payoffs Combination on Reels Hits Deduct Payouts 1 Coin 2 Coins Payout% 7B 7B 7B 2 1 1 2 -0- 400 2,000 1.
In the second column, labeled " on Reels" are the number of symbols on each reel.
For example, for the combination 5B 5B 5B, there are five 5B symbols on the first reel, 4 on the second reel and 4 on the third reel.
Following this same combination of symbols across the table, the next column shows the total number of winning combinations slot machine math Hits.
The next column, labeled Deduct shows the number of times that a symbol is used in computing a different payoff, with the same symbol used.
It is deducted so that we don't count the same symbol twice.
You will notice that in the row for AB AB AB we deduct 811 hits from the total number of hits for this combination of symbols.
This is done 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 slot machine math a total of 811 Any Bar hits which have been included in different payoff combinations.
The Payout Columns are broken down into payouts for one and two coins played.
The amounts in these columns have been computed by multiplying the payoff for each combination of symbols, as shown in Table 8, times the number of Hits for that combination.
Returning to the 5B 5B 5B combination, we compute the payouts for one coin as 80 Hits x 50 coins for a payout of 4,000 with one coin played.
If we add up the total number of hits, we have a total of 9,131 hits, before deducting overlapping symbols.
Deducting 811 for overlaps gives us a net total of 8,320 hits which will pay off on this machine.
To compute the amount the slot will retain, we divide the total number of payouts by the total number of possible combinations for: Payouts of Payouts Total C ombinations Payout Percent 1 Coin 28,499 32,768 86.
For ordinary play, the higher paying combinations of 5B 5B 5B and 7B 7B 7B, with a combined percentage of the machine payout of about 15% 1.
File this fact away for future reference as we shall use this information as part of our basis for developing our winning slots strategy.
The above was taken from the number one slot machine system guide - Gamblers Bookcase 5901-J Wyoming NE Suite 305 Albuquerque NM 87109 � 1998 - 2013 Gamblers Bookcase � All Rights Reserved.

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