JGI:Issue 20, June 2007.

This paper presents a sample three-reel three-coin slot machine game with a bonus for three coins, and a true payback percentage of 85.6% when one or two coins are wagered and 92.5% when three coins are wagered. The player sees the winning or losing combination of three symbols on the payline as well as (a) the physical reels as they scroll by and (b) what is just above and just below the payline at the end of play. An analysis of this game shows that observing the physical reels and what is just above and just below the payline indicates that the slot machine would lose money, and thus the player would make money, as the game would have a payback percentage in the range of 192%–486% if this reflected reality. The paper concludes by discussing the results of the analysis in terms of gaming regulations and problem gambling.

Keywords: slot machine, probability, randomness, virtual reels, gaming regulations, problem gambling

The payback percentage of a slot machine is determined by a computer program inside the slot machine. The underlying algorithms that the computer uses to create a slot machine game have been described by Turner and Horbay (2004) in their paper directed toward counsellors who treat and researchers who study problem gambling. The algorithms are also documented in articles in other disciplines, such as the gaming industry papers by Locke (2001) and Wilson (2003, 2004a, 2004b, 2004c, 2004d, 2004e, 2004f) and by a senior executive from an independent gaming lab (Maida, 1997). The algorithms are based on a recently expired patent (Telnaes, 1984).

The payback percentage of a slot machine game cannot be determined by examining (a) the symbols on the physical reels in the slot machine or (b) what is displayed just above or just below the payline in the payline window at the end of a play. The purpose of this paper is to use a sample slot machine game to determine the difference between the true payback percentage, as determined by the computer, and the payback percentage as indicated (a) on the physical reels and (b) by what is displayed just above or just below the payline in the payline window at the end of a play.

The difference between the true payback percentage and the payback percentage as indicated on the physical reels will be termed the physical reel distortion factor (PRDF). The difference between the true payback percentage and what the player sees just above and just below the payline in the payline window will be referred to as the payline window distortion factor above/below (PWDFa and PWDFb, respectively).

The paper is written to help problem gambling researchers better understand how slot machines can be random and yet guarantee that the physical reel distortion and the payline window distortions do exist.

To do this analysis, a slot machine pay table is needed. The manufacturers of slot machines and the jurisdictions in which they are located do not make the pay tables publicly available. Thus, a sample slot machine pay table detailed by Wilson's seven articles in Slot Tech Magazine is used (Wilson , 2003, 2004a, 2004b, 2004c, 2004d, 2004e, 2004f). It is a three-reel three-coin slot machine with a bonus for the maximum bet of three coins. Although there are many different slot machine games available on the market, Wilson chose to document a simple three-reel three-coin machine to keep the calculations "simple and easy" (Wilson, 2003, p. 12).

Using the sample slot machine from Wilson, the first section of this paper shows the calculations that determine the payback percentage based on the physical reels, while the second section shows the true payback percentage as determined by the computer. In the third section, an analysis is done on the difference between the true payback percentage and what appears just above and just below the payline in the payline window. The fourth section discusses the distortions as they relate to gaming regulations and problem gambling.

Until the mid-1980s, the true payback percentage on a slot machine could be calculated using the physical reels. Older, mechanical slot machines were built so that each symbol on each reel had an equal chance of occurring on the payline. The reels commonly had 22 stops, so the total number of reel combinations on the payline in a three-reel mechanical slot machine was 10,648 (22 × 22 × 22).

When computers were introduced into slot machines, the computer randomly controlled the outcome with an equivalent number of combinations as the mechanical slot machines had, so that a slot machine with 22 stops per reel would continue to have 10,648 reel combinations on the payline. The technique the computer used for doing this was patented by Saxton (1978) and used a straightforward mapping of random numbers to the 22 stops.

In this section, the payback percentage of a sample slot machine game is calculated using the physical reels as though the physical reels represented the odds as they did in the older, mechanical slot machines.

The game Wilson designed is a three-coin, three-reel slot machine with 22 stopping positions per reel and a bonus for the jackpot on a maximum bet of three coins. On each reel, half of the stops are blank and half are symbols. The layout of the three physical reels is shown in Table 1. Note that in this slot machine the layout of all three physical reels is the same. This is found among some slot machines, but in others the layouts of the three physical reels are different from one another. The calculations and descriptions in this paper apply equally to slot machines in which all three physical reels are the same and slot machines in which the physical reels are different from one another, so the PRDF and the PWDF calculations will be the same in both instances.

#	Symbol
1	Double Bar
2	–
3	Single 7
4	–
5	Double Bar
6	–
7	Double 7
8	–
9	Triple Bar
10	–
11	Single 7
12	–
13	Single Bar
14	–
15	Single 7
16	–
17	Single Bar
18	–
19	Double 7
20	–
21	Triple Bar
22	–

For this sample slot machine, the pay table in Table 2 contains the pay glass information—the winning combinations and what they pay. Table 2 shows, for example, that three double 7 symbols on the payline pays 500 credits if one coin is wagered, 1,000

credits if two coins are wagered, and a bonus jackpot of 6,000 credits if three coins are wagered. Three double 7 symbols is the only winning combination with a bonus for the third coin. All other winning combinations are linear payouts, with two and three coins paying two and three times as much as one coin would.

Table 2 shows what the pay table for this slot machine would be if the physical reels were used to determine the true odds. The calculations for the top three winning combinations will be discussed here.

There are eight combinations of three double 7 symbols on the payline because there are two double 7 symbols on each reel (2 × 2 × 2). Thus, the chance of getting any combination of three double 7 symbols is 8 out of 10,648, the total number of reel combinations. There are three single 7 symbols on each reel, thus there are 27 combinations of three single 7 symbols on the payline (3 × 3 × 3) out of 10,648 total reel combinations.

Any three 7s is a winning combination. There are five 7s on each reel (two double 7 symbols and three single 7 symbols), giving 125 reel combinations of any three 7s (5 × 5 × 5) out of 10,648 total reel combinations. However, slot machines pay only the highest amount for any combination of 7s, so we have to subtract from the 125 combinations the eight occurrences of three double 7 symbols on the payline and the 27 occurrences of three single 7 symbols on the payline, leaving 90 combinations (out of 10,648 total reel combinations) that would pay for any three 7s (125 – 8 – 27).

The payback percentage is the average amount that is paid on each play. For example, a payback percentage of 90.0% means that, on average, the slot machine pays out 90.0% of the amount that was wagered. Table 2 shows the calculation of the payback percentage as if physical reels were used to determine the payback percentage. With 22 stops, the total number of reel combinations is 10,648 (22 × 22 × 22). For one coin wagered, the payback over these 10,648 reel combinations is 19,650 credits, yielding a payback percentage of 185% (19,650/10,648). The payback percentage for two coins is also 185% (39,300/21,296). For three coins, the total wagered over the 10,648 combinations is 31,944 (10,648 × 3) and the payout is 94,950, yielding a payback percentage of 297% (94,950/31,944). If the physical reels accurately reflected the outcome, the casino would lose money on this slot machine, and players, on average, would make money.

But slot machines make money. Gross gaming profits in Ontario for 2004 are reported in the Ontario Lottery and Gaming Corporation's annual report and fact sheets (OLGC 2006a, 2006b) and the Canadian Gaming News (Sack 2005a, 2005b). Using the data from OLGC and Sack's data for eight gaming facilities, we can calculate that the average annual gross profit per slot machine at these eight gaming facilities is $198,828 (with a range from $80,300 to $350,765), yielding an annual gross profit of $1,179,845,352 from the 5,934 slot machines in these eight facilities.