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Russell Hunter 
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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 casino.  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 other 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 symbols 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 to 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.  For a three reel machine with twenty stops per reel, we have 20 x 20 x 20 = 8,000 combinations of slot symbols.

            If a jackpot offered on this machine pays on 7 7 7 and only one 7 symbol is on each reel, then the probability of hitting this jackpot is 1/20 x 1/20 x 1/20 or one in 8,000.  If two 7 symbols are on one reel, then our calculation is 2/20 x 1/20 x 1/20 for a probability of 1/4,000 of hitting the jackpot.

            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 as 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 a greater effect on the probabilities than the number of symbols per reel.  If we compare a machine with 32 stops per reel and 3 reels, with a 22 stop per reel machine and 4 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 5 reel machine with 32 stops per reel, we find over 33 million combinations!

            Every slot machine has a predetermined payout percentage.  When you hear things like "our slots pay back 98.3%" this means that over the long-term for every dollar inserted in the machine, it will return 98.3 cents.  Conversely, we could state that as for every dollar played, the casino will retain 1.7 cents.  These percentages only hold true over very long-term play consisting of hundreds of thousands or even millions of plays.

            Many people misinterpret these percentages and think that if they play with  $100.00 on a 98.3% payback machine that they can only lose $1.70.  There are a couple of things wrong with this line of thinking.  First, theoretical percentages will be attained only over long periods of play.  Over a few dozen, or even a few hundred rolls, the payback percentage will vary greatly.  Secondly, if a person brings $100.00 for slot play, he or she usually will not limit his or her play to inserting this amount of money into the machine only one time.  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 how it was possible for a 98.3% slot to take all of their money.

            The answer is that the casino continues to extract its percentage on every coin inserted into 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.  This table compares slot hold percentages of from two percent to fifteen percent for ten rounds of play, starting with $100.

          Table 6.  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

            With a 15% casino hold, there is only $85 left after one round of play, and after ten rounds the $100 has been reduced to only $20.  If we continue to play the 15% hold game, after twenty rounds we will be down to about $4.  We can see the power of a hold rate of 15%.  

            If we contrast this with the 2% hold rate, we see that after ten games we still retain $81.  Even though we have gradually lost some money on 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.S. is in Downtown Las Vegas, with an average payback of 95.4%.  The average for all Nevada casinos is 95.06%.  The next best place to play is in Colorado, with an average rate of 93.06%.

            Moving east, we find that the Illinois river boats offer the third best choice at 92.3%, followed by Foxwoods in Connecticut with 91.7%, the Iowa's river boats at 91.63%, and in last place the Atlantic City casinos with an average payback of 91.23%.

            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 Nevada, the win rates average about 95% 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 89.85% win rate.

            If we decide to limit our serious playing to Nevada, and stay with $1 slots, we will find our best action in Reno.  If we are a quarter player, then Downtown Las Vegas is the best deal in the U.S.

                     Table 7. Casino Win Percentages

Location

   5

Win%

  25

Win%

    $1

Win%

   $5

Win%

Total

Win%

Nevada

Las Vegas- Downtown

91.3

95.8

95.1

96.7

95.4

Las Vegas- Strip

88.9

94.1

95.3

96.3

94.7

Laughlin

87.9

94.6

95.9

96.7

94.9

Reno

91.8

94.1

96.2

97.4

95.4

Lake Tahoe

90.3

93.4

95.6

96.8

94.9

Average

90.04

94.40

95.62

96.78

95.06

Atlantic City

Average

89.95

90.91

91.63

94.18

91.25

Connecticut

Foxwoods

90.5

91.3

91.9

94.4

91.7

Illinois

Average

86.3

90.69

93.06

94.37

92.30

Iowa

Average

92.50

90.90

92.60

NA

91.63

Colorado

Average

89.33

92.87

94.00

95.17

93.06

            Examining an individual slot machine in some detail will further illuminate how slots are programmed to pay off.  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.

Table 8.  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 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.  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

 

            With 3 reels of 32 symbols each, we have a total of 32,768 combinations of symbols possible (32 x 32 x 32 = 32,768).  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.

                Table  10.  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

  1.4%

5B 5B 5B

 5  4  4

    80

 -0-

 4,000

 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

 

 

 

 

           

            The first column shows each winning combination.  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 (called Hits).  For the 5B 5B 5B combination, we have 5 x 4 x 4 = 80 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 for 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.  When two coins are played and this combination shows, we compute the payout as 80 Hits x 150 coins = 12,000 coins, reflecting the bonus payoff.

            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 Combinations

Payout Percent

1 Coin

28,499

32,768

86.97%

2 Coins

62,198

65,536

94.90%

            If you will look at the last column in Table 10, showing the Payout Percentages you will notice that almost 85% of the payouts occur on the lower paying combinations which pay out 1, 5 and 10 coins with only one coin played (18.2% + 40.8% + 25.6% = 84.6%).  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% (1.4% + 14.0%), are much less relevant to assessing how will this machine will pay for short-term play.  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 - Super Slots!

 

 

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