Five-Line Australian Reels Slot Machine Math

Introduction

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.

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.

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

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

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).

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.