Slot Machine Appendix 6 - Red White & Blue Analysis

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In the past I feared to show a concrete example on this page, not wanting to violate copyright laws. However, in the Netherlands, information on the reel stripping and probabilities of each win are sometimes posted on small cards on the machine. A fan of the site sent me the information provided on an IGT "Red White & Blue" machine.

First, let me present the pay table. For those who are not familiar with the game, a 3-bar is in blue, a 2-bar is in white, and a 1-bar is in red.

Pay Table for Red White & Blue

Win 1 Coin 2 Coins 3 Coins
Red 7, white 7, blue 7 2400 4800 10000
Red 7, red 7, red 7 1199 2400 5000
White 7, white 7, white 7 200 400 600
Blue 7, blue 7, blue 7 150 300 450
Any 3 sevens 80 160 240
1 bar, 2 bar, 3 bar 50 100 150
3 bar, 3 bar, 3 bar 40 80 120
2 bar, 2 bar, 2 bar 25 50 75
Any red, any white, any blue 20 40 60
1 bar, 1 bar, 1 bar 10 20 30
Any 3 bars 5 10 15
Any 3 reds 2 4 6
Any 3 white 2 4 6
Any 3 blues 2 4 6
Blank, blank, blank 1 2 3

After the player makes a bet and presses the spin button, the machine selects three random numbers, one for each reel. These are chosen from a random number generator that is constantly drawing random numbers at a rate of thousands per second. The numbers chosen at the moment the play is initiated are the ones used to determine the final outcome. In other words the outcome is predestined the moment you press the spin button.

In the case of Red White & Blue, each random number has 64 equally likely outcomes. Each random number is mapped to a stop position on what is called a "Lookup Table." The following is my estimate of what the Red White & Blue lookup table looks like, based on the total number of stops per symbol per reel, provided to me.

Lookup Table for Red White & Blue

Stop Number Reel 1 Reel 2 Reel 3
1 2 bar 2 bar 2 bar
2 2 bar 2 bar 2 bar
3 2 bar blank 2 bar
4 blank blank blank
5 blank 3 bar blank
6 3 bar 3 bar 3 bar
7 3 bar blank blank
8 blank blank blank
9 blank blank blank
10 blank white 7 white 7
11 white 7 blank white 7
12 white 7 blank white 7
13 white 7 blank white 7
14 white 7 1 bar white 7
15 white 7 1 bar white 7
16 white 7 1 bar white 7
17 blank 1 bar blank
18 blank blank blank
19 blank blank blank
20 1 bar blank 1 bar
21 1 bar blue 7 1 bar
22 1 bar blue 7 1 bar
23 blank blue 7 1 bar
24 blank blue 7 1 bar
25 blank blue 7 blank
26 blue 7 blue 7 blank
27 blue 7 blue 7 blank
28 blue 7 blank blue 7
29 blue 7 blank blank
30 blue 7 blank blank
31 blue 7 2 bar blank
32 blank 2 bar 2 bar
33 blank blank 2 bar
34 blank blank 2 bar
35 2 bar 3 bar blank
36 2 bar 3 bar blank
37 blank blank 3 bar
38 blank blank blank
39 3 bar blank blank
40 blank blank blank
41 blank blank blank
42 blank red 7 blank
43 blank red 7 red 7
44 blank red 7 blank
45 red 7 blank blank
46 blank blank blank
47 blank blank blank
48 blank blank blank
49 blank blank 3 bar
50 blank 3 bar 3 bar
51 3 bar 3 bar 3 bar
52 3 bar 3 bar blank
53 3 bar blank blank
54 blank blank 2 bar
55 blank 2 bar 2 bar
56 2 bar 2 bar 2 bar
57 2 bar blank blank
58 blank blank blank
59 blank 1 bar 1 bar
60 1 bar 1 bar 1 bar
61 1 bar 1 bar 1 bar
62 1 bar 1 bar 1 bar
63 blank blank blank
64 blank blank blank

Immediately after the random numbers are chosen they will get mapped to a stop on the machine via the Lookup Table, the outcome will be scored, and the player paid if he won anything. For example, if the random numbers chosen were 26, 8, and 43 the player would get Blue 7, Blank, Red 7.

Note how there are clusters of the same symbol in a row. For example, stops 60 to 62 on reel 1 all are mapped to a 1 bar symbol. These will all be directed to the same 1 bar symbol on the actual reel. There are exactly 22 groups of like symbols on each reel, which is the standard number of stops on an electro-mechanical three-reel slot machine, known as a "Stepper Slot."

Also note that only stop 45 on reel 1 is mapped to the red 7 symbol. However the blanks above and below it have five positions each. This causes the often seen near miss effect, where the reel stops directly above or below the highest paying symbol. Many people have written to me, claiming that this near miss effect is against the law. My slot machine Appendix 1 proves otherwise.

To calculate the return of the machine, add up the number of positions for each symbol on each reel, to get the total symbol weightings. The following table shows these totals. This table is what was indicated on the Red White & Blue machine in the Netherlands. The Lookup Table above, was reverse engineered to produce these total weights.

Total Weights for Red White & Blue
Stop Number Reel 1 Reel 2 Reel 3
Red 7 1 3 1
White 7 6 1 7
Blue 7 6 7 1
3 bar 6 7 5
2 bar 7 6 9
1 bar 6 8 9
blank 32 32 32
Total 64 64 64

With the weighting of each symbol and reel known, it is just take simple math to calculate the return. The following table shows the win, number of combinations, probability, and contribution to the return for all possible events. For example, the table above shows the number of white 7's are 6, 1, and 7, for reels 1, 2, and 3 respectively. The total number of winning combinations for three white sevens is thus 6 × 1 × 7 = 42.

Return Table for Red White & Blue

Win Pays Combinations Probability Return
Red 7, white 7, blue 7 2400 1 0.000004 0.009155
Red 7, red 7, red 7 1199 3 0.000011 0.013721
White 7, white 7, white 7 200 42 0.000160 0.032043
Blue 7, blue 7, blue 7 150 42 0.000160 0.024033
Any 3 sevens 80 1199 0.004574 0.365906
1 bar, 2 bar, 3 bar 50 180 0.000687 0.034332
3 bar, 3 bar, 3 bar 40 210 0.000801 0.032043
2 bar, 2 bar, 2 bar 25 378 0.001442 0.036049
Any red, any white, any blue 20 113 0.000431 0.008621
1 bar, 1 bar, 1 bar 10 432 0.001648 0.016479
Any 3 bars 5 7977 0.030430 0.152149
Any 3 reds 2 335 0.001278 0.002556
Any 3 white 2 1036 0.003952 0.007904
Any 3 blues 2 756 0.002884 0.005768
Blank, blank, blank 1 32768 0.125000 0.125000
All other 0 216672 0.826538 0.000000
Total 262144 1.000000 0.865761

The lower right cell shows a return of 86.58%. That means that for every dollar bet at the one coin level, the player will get back 86.58 cents, on average. Doing the same table for two coins results in a return of 86.58% as well, and three coins has a higher return of 87.47%, due to the disproportionately high win on the top two pays. The standard deviation is 9.03 for 1 or 2 coins, and 10.80 for 3 coins.

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