A study of game mathematics can be valuable when considering which games to place on your floor, and it is necessary when determining the variance those games will have once your customers begin to play them. This month we’ll take a look at hit frequency and study the basics of this calculation.
First we must define two terms. The first is the cycle, which is the total number of unique combinations that exist in a game. It is determined by multiplying the number of symbol stops on each reel together. The second is hit. A hit is a game outcome that results in a win of some form, such as a standard payout or a bonus game trigger. Hit frequency refers to an expression of how frequently the winner will receive a hit; this is generally expressed as a percentage. Figure 1 shows the calculation for hit frequency. A hit frequency of 10 percent means that the player will win something in one game out of every 10, on average.
For example, let’s say there are 32,768 possible combinations in the game (32 symbols on each of the three reels). This is a very small cycle size, but for the purposes of illustration, it makes the math simpler when calculating hit frequency. In our hypothetical game, 4,438 spins result in a hit. Dividing the hits by the
cycle gives us 0.135437, which we multiply by 100 in order to express it as a percentage. This game will have a win 13.54 percent of the time.
Note that the hit frequency is an average based upon the complete cycle. If you were to flip a coin 100 times, you may not receive exactly 50 heads and 50 tails, but will likely end up with numbers close to that. Since each flip is random and the coin does not remember previous results, you could get 49 heads and 51 tails, 53 heads and 47 tails, etc. Likewise, the game will not play to the exact hit frequency values in the short term. Even with a 50 percent hit frequency, the player will likely not win half of the time. They may have a run where they win a number of games in a row, and they may also experience a run where they go for several games without winning anything. Since the hit frequency uses the cycle as a basis of its calculation, the hits depend upon the cycle, or the long-term experience, of the game. In the short term, the player will experience wild variances either in their favor or against them. That being said, however, the hit frequency can be used as a general guide for the short-term experience.
A game with a 10 percent hit frequency is more likely to experience a dry spell than a game with a 50 percent hit frequency. Although short-term variances will affect these values, the lower hit frequency game is less likely to experience a run with numerous wins. This can be used to determine which games suit your player base. For a high-denomination game where the players will not generally be playing very long (such as a $100 game), it makes sense to have a high hit frequency. In this case, the payout amount can be reduced without an adverse effect. For a low denomination game, a lower hit frequency can be implemented. Players on a penny slot will generally have a larger bankroll (in credits) and can therefore cover the dry spells that result from a lower hit frequency configuration.
Using hit frequency as a benchmark is a good idea when considering new games for your floor. However, the hit frequency of the game is not the only application for this formula. We can also look at the hit frequency of any particular win—such as a bonus round or jackpot—as well as groupings of hits.
In Figure 2, we see a portion of a PAR sheet’s symbol table that shows the number of symbols on each of the reels. If we are interested in knowing the frequency of jackpots, then we simply determine how many jackpot combinations there are and divide that number by the cycle. The table shows two symbols per reel for a total of eight combinations (2 x 2 x 2). The hit frequency is 0.02 percent (8 / 32,768 x 100). We could also determine how many times the players will receive a hit for any combination of 7s (jackpot, three red 7s or mixed 7s) by adding up the individual symbol counts for each reel then multiplying them together. On the first reel we have two jackpot symbols and five red 7s. Add these together to get seven of the symbols on the first reel. Repeat this process for each reel then multiply the numbers together. There are 252 red 7 combinations in the game (7 x 6 x 6), and the hit frequency is 0.77 percent (252 / 32,768 x 100). This means that the player will receive a payout for some combinations of 7s just under 1 percent of the time.
We can use this method for any symbols on the reel. For example, there are 13 bar symbols on reel one (three triple bars, five double bars and five single bars), 13 on reel two and 10 on reel three. Multiply these values together to determine that the game has 1,690 combinations of bar symbols. Calculating 1,690 / 32,768 x 100 gives us a hit frequency of 5.16 percent. The player will receive some payment for bars (mixed bars, three single bars, three double bars or three triple bars) in approximately 5 percent of the games.
We may want to further dissect the games to determine how wins are distributed within all of the hits. By doing this, we are basing our calculations only at the total group of wins. For example, in our hypothetical game in Figure 1, we know that there are 4,438 hits per cycle. If we calculate the number of pays for 7s (in any combination), we know that there are 252 instances where the player wins for some combination of 7s. If we use 4,438 as the divisor of our formula, then we are not looking at the entire game but at the entire hits. The resulting value of 252 / 4,438 is 0.05678, or 5.68 percent. This tells us that, of all the possible wins, the player will experience just over 5 percent from some form of 7s. For bars the value is 1,690 / 4,438, which is 0.38080, or 38.08 percent. When the player wins something, over one-third of the time this will be a win from various bar symbol pays. We can use varying values to determine the hit frequency of anything we want. It does not have to be just the game hit frequency. [Note: When considering any calculations and studying them as percentages, each value has to add up to 100. When we look at game hit frequency, the percentage of wins and the percentage of losses must add up to 100. When considering only hits and looking at individual percentages, such as the percentage of payouts for 7s, each individual payout must add up to 100 percent. Figure 3 illustrates this.]
You could even consider non-winning games. Suppose, for example, you wish to determine the number of games where three 7s appear in the reel glass but have not lined up for a resulting pay. First, add up the instances where three 7s appear in the reel window. To do this, locate each symbol on the reel and add up the blanks before and after. Figure 4 shows how to do this. You cannot use the symbol table for this calculation, as it does not differentiate between a blank symbol next to a 7 and one next to any other symbol. Look at the reel-strip listing and record the blank stops next to the 7s. Add up these blanks and the 7 symbol stops for each reel. In our sample game, we have 19 stops on the first reel, 17 on the second and 20 on the third. Multiplying this together tells us that there are 6,460 possible combinations where three 7s appear within the reel window. It is paramount to remember, however, that this includes all instances where three 7s are visible. In other words, this includes all combinations where three 7s appear on the payline. But for our calculation, we wish to determine when three 7s appear but do not result in a payout. Therefore, we must subtract the instances where three paying 7s appear, which we calculated earlier. If we subtract those 252 stops from the 6,460 possibilities, we determine that there are 6,208 outcomes where three non-paying 7 symbols appear in the reel window. Using our handy hit frequency formula, we learn that three 7s appear in the window in just under 20 percent of the games (6,208 / 32,768 x 100= 18.95 percent).
The appearance of “near misses” like this has been shown to be exciting for the player, but if they occur too often, the players feel that the game is just teasing them. A study of “almost won” and “did win” occurrences will allow you to see if this game will fit with your customer base.
By studying the higher paying amounts, you can begin to understand the variance potential of the machine. If there are numerous high-paying awards, then the machine has a potential for a large variance in payout. There is nothing wrong with this, but it does help to know this information before committing to the game. Such a game will not cause concern when your accounting figures show the game operating several percentage points outside of the stated theoretical hold.
Hit frequency calculations allow you to learn a lot about how a game pays and plays. Will it perform well? Will your customers like the overall “feel” of the game? Will it be financially feasible for you to operate? Does it offer payouts that will create satisfaction and game loyalty for your patrons, or will it discourage future play? A comprehensive study of hit frequency allows you to delve into the machine’s personality and understand some of the subtleties of its game-play patterns.
From here, we can progress into more advanced study. Next month, we’ll continue this analysis by addressing whether it is normal for a certain payout that should statistically occur every (n) games to not occur. We’ll study a formula that will give you the probability of a particular hit (e.g., a jackpot) occurring—or not occurring—within the “average” payout period.