Derby: The electronic horse racing game

Last update: October 25, 2002

Analysis of 21 Bell Slot Machine

This appendix attempts to show how return for the 21 Bell Slot Machine is calculated.

Table 1 shows the frequency of each symbol(s) on each reel. Each stop is equally likely.

Table 1: Reel Strips
Stop	Reel 1	Reel 2	Reel 3
1	orange	cherry	bell
2	melon	plum	orange
3	plum	cherry	plum
4	cherry	7 and orange	bell
5	plum	cherry	orange
6	orange	bell	lemon
7	7	plum and bar	bell
8	bell and bar	bell	melon and orange
9	orange	cherry	bell
10	cherry	orange	plum
11	bar	bell	lemon
12	plum	melon and orange	bell
13	orange	plum	plum
14	plum	bell	bell
15	melon	cherry	7 and bar
16	plum	bar	lemon
17	orange	orange	bell
18	plum	cherry	melon and orange
19	bar	bell	bell
20	plum	melon and orange	lemon

Table 2 shows the frequency of each symbol on each reel.

Table 2: Symbol Frequency
Symbol	Reel 1	Reel 2	Reel 3
7	1	1	1
Bar	3	2	1
Melon	2	2	2
Bell	1	5	8
Plum	7	3	3
Orange	5	5	4
Cherry	2	6	0
Lemon	0	0	4

Table 3 shows the number of combinations of each paying combination (probability combinations) and the expected return for each paying combination over all possible outcomes (return combinations).

Table 3: Probability and Return Combinations
Win	Pays	Probability Combinations	Return Combinations
3 sevens	200	1	200
3 bars	100	6	600
3 melons	100	8	800
melon, melon, bar	100	4	400
3 bells	18	40	720
bell, bell, bar	18	5	90
3 plums	14	63	882
plum, plum, bar	14	21	294
3 oranges	10	100	1000
orange, orange, bar	10	25	250
cherry, cherry, any	5	240	1200
cherry, any, any	2	560	1120
Total		1073	7556

The bottom row of table 3 shows 1073 probability combinations and 7556 return combinations. With 20 stops on each reel the total combinations is 20³=8000. So the probability of any win (hit frequency) is 1073/8000 = 13.41%. The machine's expected return is 7556/8000 = 94.45%.

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