Last Updated: December 21, 2008

Five-Line Australian Reels Slot Machine Math

This page shows how my Five-Line Australian Reels slot machine was designed. Everything you need to know about the return can be found in the pay table and reel strips. Here is the pay table:


Click image to play.

Australian Reels — Pay Table

Win 1 coin 2 coins 3 coins
Three flags 2500 5000 8000
Blue 7, white 7, red 7 (in order) 200 400 600
Three blue 7's 80 160 240
Three white 7's 60 120 180
Three red 7's 40 80 120
Three kangaroos 20 40 60
Three platypuses 15 30 45
Three echidnas 10 20 30
Any three 7's 2 4 6
Any three kangaroo,platypus,echidna 1 2 3

The next table shows the reel strips. The game picks 3 random numbers from 1 to 64, one for each reel. Then it maps each number to the corresponding position on the reels. This will decide the symbol in the middle row. The symbols on the top and bottom lines will be above and below the randomly chosen stops on the reel strips.

Australian Reels — Reel Strips

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

The first step to calculating the return is to count the number of times each symbol occurs on each reel.

Australian Reels — Symbol Count

Symbol Reel 1 Reel 2 Reel 3
Flag 1 1 1
Blue 7 2 8 8
White 7 11 2 9
Red 7 13 12 2
Kangaroo 16 14 5
Platypus 15 6 19
Echidna 6 21 20
Total 64 64 64

Grouping symbols into groups makes the math easier. Here are pertinent groups to the math.

Australian Reels — Symbol Group Count

Symbol Reel 1 Reel 2 Reel 3
Blue 7 or Flag 3 9 9
White 7 or Flag 12 3 10
Red 7 or Flag 14 13 3
Kangaroo or Flag 17 15 6
Platypus or Flag 16 7 20
Echidna or Flag 7 22 21
Any 7 or Flag 27 23 20
Any three Kangaroo, Platypus, Echidna, or Flag 38 42 45

Next comes the fun part, calculating the number of combinations for each win in the pay table.

  • Three flags = 1 × 1 × 1 = 1.

  • Blue 7, white 7, red 7 = 3 × 3 × 3 - 1 = 26. We subtract 1 because I included the flag in each group. If the player got three flags he would win the jackpot instead.

  • Three blue 7's = pr(blue 7 or flag on reel 1) × pr(blue 7 or flag on reel 2) × pr(blue 7 or flag on reel 3) - pr(flag on reel 1) × pr(flag on reel 2) × pr(flag on reel 3) - pr(blue 7 on reel 1) × pr(flag on reel 2) × pr(flag on reel 3) = 3 × 9 × 9 - 1 × 1 × 1 - 2 × 1 × 1 = 240. Here we multiply the weights for a blue 7 or flag for each reel, and then subtract for the higher paying combnations of three flags, and blue7-flag-flag, which can form the higher paying blue7-white7-red7.

  • Three white 7's = pr(white 7 or flag on reel 1) × pr(white 7 or flag on reel 2) × pr(white 7 or flag on reel 3) - pr(flag on reel 1) × pr(flag on reel 2) × pr(flag on reel 3) - pr(flag on reel 1) × pr(white 7 on reel 2) × pr(flag on reel 3) = 12 × 3 × 10 - 1 × 1 × 1 - 1 × 2 × 1 = 357. Here we multiply the weights for a white 7 or flag for each reel, and then subtract for the higher paying combnations of three flags, and flag-white7-flag, which can form the higher paying blue7-white7-red7.

  • Three red 7's = pr(red 7 or flag on reel 1) × pr(red 7 or flag on reel 2) × pr(red 7 or flag on reel 3) - pr(flag on reel 1) × pr(flag on reel 2) × pr(flag on reel 3) - pr(flag on reel 1) × pr(flag on reel 2) × pr(red 7 on reel 3) = 14 × 13 × 3 - 1 × 1 × 1 - 1 × 1 × 2 = 543. Here we multiply the weights for a red 7 or flag for each reel, and then subtract for the higher paying combnations of three flags, and flag-flag-red7, which can form the higher paying blue7-white7-red7.

  • Any three 7's = 27 × 23 × 20 - (1+26+242+359+545) = 11247. Here we subtract out all the higher pays of three of a kind.

  • Three kangaroos = 17 × 15 × 6 - 1 × 1 × 1 = 1529.

  • Three platypuses = 16 × 7 × 20 - 1 × 1 × 1 = 2239.

  • Three echidnas = 7 × 22 × 21 - 1 × 1 × 1 = 3233.

  • Any three kangaroo,platypus,echidna= 38 × 42 × 45 - (1+1529+2239+3233) = 64818. Again, we subtract out higher pays, consisting of three of the same animal or flag.

The next table shows for each win, with one coin bet: what it pays, combinations, probability (combinations divided by 643, and return (product of pays and probability). The lower right cell shows a return of 96.973%.

With 3 coins bet the return per coin bet for three flags is 8000/3=2666.67. That increases the win for three flags by 166.667. That increases the return by (1/643)×166.667 = 0.000636. The return for 3 coins bet is thus 96.840% + 0.064% = 96.903% (total not equal to sum of parts due to rounding).

Australian Reels — Summary

Win Pays Combinations Probability Return
Three flags 2500 1 0.000004 0.009537
Blue 7, white 7, red 7 200 26 0.000099 0.019836
Three blue 7's 80 240 0.000916 0.073242
Three white 7's 60 357 0.001362 0.081711
Three red 7's 40 543 0.002071 0.082855
Any three 7's 2 11253 0.042927 0.085854
Three kangaroos 20 1529 0.005833 0.116653
Three platypuses 15 2239 0.008541 0.128117
Three echidnas 10 3233 0.012333 0.123329
Any three kangaroo,platypus,echidna 1 64818 0.247261 0.247261
Total 84238 0.321342 0.968395

This design is for educational purposes only. It is my design. Permission is NOT given to steal it to make a real money game. For such permission, please contact me.

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