Last month we examined the foundation of hit frequency, an indication of how frequently a player will experience a win. While we tend to think of hit frequency as a component of the entire game cycle, it can also relate to individual payouts. Grouping symbols together allows us to study the occurrence and impact these symbols have on the play and pay of the game. We can also study certain pays as a component of other pays. For example, we can determine how frequently bar payments occur as well as investigating the frequency of mixed bar payouts within this grouping. Hit frequency is a valuable component of game play and can have a major impact on both game satisfaction for the players and the potential volatility of the game.
Once we are familiar with the calculations for hit frequency, we can take these basic theories and apply them for more advanced study. Not only will this study allow you to examine the game’s play, but you will also be able to mathematically show whether the game is playing within statistical norms. This month we continue with a study of game play and an analysis of game statistics. Next month we will look at a more advanced formula that will allow us to provide mathematical quantification of game statistics and provide a valuable tool for your game analysis, bringing everything together.
Is This Game Working?
Let’s begin by studying a fictitious game on a casino floor. A single-game progressive, Crackpot Jackpot, has been on your floor for two years. The progressive jackpot has never been hit and a customer has complained that the game is either not playing properly or has been “fixed” so that it doesn’t pay out. Your boss has given you the task of studying this game and providing proof that the game is playing properly. A study of the hit frequency will help you verify the game’s proper operation, because we are really asking if the frequency of this pay (or lack of pay) is within statistical norms.
First you check the EEPROM or flash memory in the machine and verify that it matches the approved game personality. Once this has been done, you have a pretty good idea that the game is operating correctly. If the software and ROM are correct, it is unlikely that the machine is malfunctioning. At a minimum, it will show that there has been no tampering with the machine’s programming. With this knowledge in hand, let’s continue our forensic investigation.
We can find some of the basic information that we require on the game’s PAR sheet (see Figure 1), but we’ll also need to check the machine for game play information. This could come from hard meters within the game or a slot report generated from your system (see Figure 2). This will provide actual play results that can be compared to the theoretical play information derived from the PAR sheet.
The two sets of numbers will not be identical for several reasons. First of all, the PAR sheet is theoretical in nature. It shows the values that work out on paper—not on your floor. Second, each game is random, resulting in less-than-predictable results. Third, the game does not remember outcomes from previous games. For example, one combination may occur twice before another combination occurs once. For Crackpot Jackpot, we know that the top jackpot has not been hit, and this is important.
Now we have all the information we need to continue our study.
Is It Plausible?
Our first task is to examine the numbers to see if they seem correct. You can often determine a lot about a machine just by taking a cursory look at the numbers without even analyzing the data.
Suppose, for example, that the coin-in value for a three-coin game was 870,894, and the games played counter read 112,394. Dividing the coin-in by the games played gives us a value of 7.75 coins per game (870,894 / 112,394 = 7.75). Since this is a three-coin game, this figure can’t possibly be correct. The data, therefore, is erroneous.
Before you spend time conducting detailed studies of your numbers, try to determine if those numbers seem correct. There could be a data problem or any of many other circumstances that result in invalid numbers being reported. If the numbers “look” right, then continue. If the data is clearly invalid, like in our example above, there is no point in continuing the investigation. In these cases, your next order of business would be to analyze the numbers from as many sources as possible, trying to identify errors and determine the correct values.
For Crackpot Jackpot, the coin-in divided by the games played is 870,894 / 302,394 = 2.88. This seems reasonable since it is a three-coin game. You can compare it to other games on your floor with the same maximum-credit wager and the same denomination. As this is a progressive game, it would seem reasonable that most people would wager maximum credits, as they hope to hit the jackpot. However, not everyone will do so, so we should expect this number to be slightly lower than the maximum credit potential.
The payout can be confirmed by dividing the coin-out by the coin-in. For Crackpot Jackpot, the payout is 802,093 / 870,894 = 0.92099957, or a payout percentage of 92.10. The theoretical payout is 95 percent, giving us a difference of 2.9 percent. Because the largest payout, the jackpot, has not been paid, we should expect the actual payout to be less than the theoretical payout. This value seems correct—at least it is close to theoretical. In order to study this figure in more detail, determine the total percent of coin-out that the jackpot amount contributes and the total coin-out payment for the cycle, then divide the total coin-out for the jackpot by the coin-out for the cycle.
We don’t know all of the data, so we must calculate the missing information. We can determine the total coin-out for the cycle by multiplying the games in the cycle by the maximum number of coins that can be wagered (373,248 x 3 = 1,119,744). The jackpot pays 25,000 as the base amount (25,000 / 1,119,744 = 0.0223). The jackpot, therefore, uses 2.23 percent of the total coin-out in a cycle. If we add this amount to the actual coin-out so far, we have a pay amount of 92.1 + 2.23 = 94.33. If the jackpot had been paid, then the actual payout percentage would be 94.33, or a difference of 0.67 percent, which is incredibly close to the theoretical value.
The hold is determined by subtracting the payout from 100, or by dividing the coin-held by the coin-in. The hold for Crackpot Jackpot is (870,894 – 802,093 ) / 870,894 = 0.07900. Note that this corresponds to the payout percentage we already calculated, as 1 – 0.9210 = 0.07900. By verifying this calculation two ways you can ensure that the numbers are correct.
The report in Figure 2 shows 45,782 winning games. Divide this value by the number of games played to calculate the actual hit frequency for the game (45,782 / 302,394 = 0.1514). The theoretical game hit frequency is 17.36 percent, so our calculation of an actual hit frequency of 15.14 percent is 2.22 percent lower than theoretical. Since our payout percent is slightly lower than theoretical, the hit frequency may also be lower as well. But as long as it is within a couple of percentage points, it should be OK.
Another calculation you can perform on the values is to confirm the amount of the progressive jackpot. Use the following formula: base-amount + (coin-in x incremental-percentage) x denomination. For this machine, the progressive jackpot should currently be $25,000 + (870,894 x 0.01) x $1 = $33,708.94. This confirms that the value of the progressive meter is correct. If this matches the reported value, then the coin-in values and the incremental percentage should be correct.
To estimate the correctness of the games played, divide this value by the number of days that have passed. This occurred over two years, so we will assume that your casino is open 365 days per year. Divide 302,394 by (365 x 2) to get an average value of slightly more than 414 games per day. This equates to 17 games per hour. This value may be low, but perhaps the game is located in a low traffic area, the denomination is higher than your market prefers, or one of many other possible reasons. Compare this value to other games in the physical area of the casino, other games of the same denomination and other progressive jackpot games.
Now we move on to the situation of the progressive jackpot not occurring. First, the time that the game has been in use is irrelevant. The number of games played is more important. The cycle of this game is 373,248, and we report 302,394 games having been played. As we have not yet completed a cycle, it does not seem unreasonable that a jackpot has not been hit. This is the first thing to look at when determining if the information seems “reasonable.” While this is not a scientific evaluation of the data, it does give you an idea as to whether or not the situation could occur. A little later on we’ll be able to actually quantify this and determine whether this situation is statistically probable.
As each game is random, the jackpot could occur at any time. It is quite possible that the jackpot could hit on the first game played. It is also possible that the jackpot could hit on the 373,248th game played. Could it hit on the second cycle instead, waiting until game 746,496? Certainly it could. At this point, Crackpot Jackpot has completed 302,394 / 373,248 of a cycle, or 0.81017. With 81 percent of a cycle complete, we still have 19 percent to go. That translates to 70,854 games (373,248 – 302,394 = 70,854) remaining until one complete cycle has occurred. Based upon the average level of play, the cycle will be completed in 70,854 / 414 = 171 days, or almost six months from now.
If we were to draw a graph showing where the jackpot hit is likely to occur, it would look similar to Figure 4. It appears that as we near a full cycle, there would be an increased chance of hitting the jackpot. As we leave the cycle, we would have a decreased chance of hitting the jackpot. But does that mean that the probability of hitting the jackpot actually increases the more games we play? Or that the probability actually decreases after we’ve played past the cycle?
While Figure 4 might suggest that the chances of obtaining the jackpot increase as we approach a full cycle, it is actually the cumulative opportunity that we are dealing with. You have a fixed probability of obtaining the jackpot with every game. This does not change between the first game and game No. 373,248. As the progressive jackpot amount increases, it does not mean that the jackpot is more likely to hit. If we were to simulate this slot machine and mark the game number where the jackpot hit, we would have a graph similar to Figure 4. It would more likely be that of a standard bell-curve as shown in Figure 5. Once again, we need to think of the jackpot hit considering the life of the slot machine, not the player sitting at the game. The less time the game has been in play, the less likely the jackpot is to have occurred. While each game has the same probability, it is the cumulative play over time that means there is a greater chance of having the jackpot hit. With one game out of 373,248, there is a very slight chance of hitting the jackpot. But after the game has had 100,000 plays, then there have been 100,000 games—each with the same probability—in which the jackpot could have occurred. By the time that a full cycle has been reached, it is quite likely that the jackpot will have hit. However, it is not a certainty. This “rule of thumb” can be used to see if the game appears to be playing properly. However, you can think of this as a “gut feeling,” which does not give scientific proof.
Close, But What’s Next?
By now we have a good feeling that Crackpot Jackpot is playing correctly. The figures “look correct,” which gives us a general feel for the game, but it is not a complete investigation—really, it isn’t anything more than a cursory look at the game. I am not suggesting that you can verify a game via a hunch or a feeling, but this initial investigation can save you from doing hours of research only to later find out that your data is bad.
Next month we are going to continue our analysis with Crackpot Jackpot, delving into a formula that will provide a mathematical indication of how the game is working. Being able to actually quantify performance and provide a numeric value will give our investigation credibility and offers a scientific answer.
Oh, yeah, and we’re going to have some free software to play around with, too!