Casino Enterprise Management

Each of us has certain dreams. We might dream about winning the lottery, paying off the mortgage or going on vacation. We also have certain expectations—more realistic expectations—especially for the machines on our floors. We expect to maintain a certain hold and to obtain a desired level of daily win. Our customers have certain dreams and expectations, too. Some dream about winning enough to change their lives. Others expect to win enough to keep playing for the evening or to win the new bonus game. Part of the difficulty in our expectations is expressing them as a mathematically quantifiable value.

The study of slot machines in a mathematical sense will allow you to determine the hold you can reasonably expect to obtain. If you take a look at a machine’s PAR sheet, you will likely see the term “expected value,” or EV for short. Expected value is an actual mathematical term. In dealing with probability or statistics, the expected value of a random variable (e.g., a slot game outcome) is the integral of the random variable with respect to its probability measure. Yeah, that certainly clears things up, doesn’t it?

Let’s take a look at expected value in a simple sense and then apply it to our slot machines. First of all, the expected value is not what we expect the machine to pay. It does not refer to the value that the player is going to be awarded from the base game, a bonus game, a progressive jackpot, etc. In fact, the expected value may not even be a possible payout amount. When we speak of expected value, we refer to the amount of credits that we pay the player, on average, over the long term. This is the average paying amount that we expect them to receive.

Suppose that we roll a standard six-sided die. We could write an equation for expected value like this:
This equation shows that the average value from the roll of the die will be 3.5. We can obviously never roll a 3.5. But if we were to give the player $1 for each spot that appears, then we will pay, on average, $3.50 per roll of the die.

With any average, there must be a lower amount and a higher amount possible. The roller could get a one or a two (lower than the average), a five or a six (higher than the average), but never a 3.5. It would be a safe assumption, however, that if you had 1,000 people roll the die, at the end of the day you would have paid out $3.50 x 1,000 = $3,500. You may have actually paid out a little more or a little less, but it’s probably going to be close to $3,500. The roll of the die is, after all, a random event. And the expected value is, after all, an average amount.

Bonus Time
The same logic applies to our slot machines. Let’s examine a fictitious bonus round where the player wins a random amount. The bonus round will award one of 13 possible values. How the bonus round does this is not important to our study. The player may simply be shown the amount, or she may select items and accumulate credits until she uncovers “COLLECT.” The process is irrelevant. If she is going to be awarded 100 credits, it is mathematically the same whether she is simply told that she won 100 credits or if she picks items and is awarded 10, 20, 10, 30, 10, 15, 5 then COLLECT. In both cases, she receives 100 credits.

The chart in Figure 1 shows the possible amounts that the player can win. If she is given 10,000 credits, it will affect our hold more than if she is given 10 credits. As a result, we will likely want to pay 10,000 credits much less frequently than we pay 10 credits. To do this we assign each value a probability—the probability that this particular award will be paid during any bonus round initiation. The award probabilities must all add up to 1. A probability of 1 will always happen. That means that we will always pay an amount to the player. Figure 2 shows each award value and its associated probability. This essentially tells us how likely it is that each amount will be selected. The higher the probability, the more likely that particular amount will be won. If we wish to pay out some very large amounts, we need to be able to control the frequency with which we award them. By assigning a low probability, we can ensure that these are paid infrequently. This lessens the overall effect on our hold.

In our bonus game, the top award of 10,000 credits is assigned a probability of 0.01.  If we multiply the probability by 100, we can express the value as a percent. In this case, we have a 1 percent chance of paying the 10,000 credit top award. Or, in other words, we have a 99 percent chance of paying a different award. This means that the top award is paid out rarely, and our average payout will be significantly less than 10,000. The most probable award is 500 credits, which has a probability of 0.11, or 11 percent.

In order to calculate what we expect to pay for this bonus round, we must add another equation. We simply take the bonus award and multiply it by its associated probability. The top award of 10,000 credits multiplied by 0.01 is 100. This is the expected value for this award. It essentially takes, on average, 100 credits for each bonus game. This makes sense because, at 1 percent, it would take 100 bonus games to make up this amount. 100 credits x 100 bonus games = 10,000. In this sense, we can think of it as a type of budget. Every time the bonus round starts, we put 100 credits aside for the eventual payout of the top award. After 100 bonus games have been played, we have now “saved up” the top award. It is vitally important to remember, however, that each outcome is random. There is no actual setting aside of this amount, even though players feel that this occurs, and that is why they avoid games that have just paid out a large amount. But remember, this budget is a mathematical expression only. The top bonus award could be paid out twice in a row—it all comes down to a random outcome.

Looking at the next-highest award in Figure 2—5,000 credits—we see that it has a probability of 0.02, or 2 percent. 5,000 x 0.02 = 100. This means that every time we initiate the bonus round, we allocate 100 credits for the second amount. Note that the second amount is one-half of the top award, yet we still allocated the same amount. That is because it is paid twice as frequently. It comes up 2 percent of the time, while the top award comes up 1 percent of the time.

The bottom award—10 credits—has a probability of 0.05, or 5 percent. 10 x 0.05 = 0.50, meaning that for each bonus round, we only put one-half of a credit aside for this payout. This payout has an insignificant affect on our hold, as it is so low. It is also well below the average—or expected value—of 463 credits, as we’ll calculate below. [Note: The probability of this payout is also less than that of higher amounts. You will frequently see this in bonus games or various payout schedules in your PAR sheets. This creates a situation where the player will not win the minimum amount very often. We like to have our players win something other than the lowest possible outcome in order to increase their happiness.]
After we have calculated each of the payment amounts, we add them up. Figure 2 shows that this value is 463 credits. This is the expected value from the bonus game. Each time we initiate the bonus game, we expect it to cost us 463 credits. You will note, however, that there is no payout of 463 credits. But, on the average, this is what we are paying. This takes into account the lower payouts (e.g., 10 credits) and the higher payouts (e.g., 10,000 credits).

For the overall game mathematics, we now have a quantifiable value to tell us what our expectations from the bonus game are. This allows us to continue our studies to ascertain how volatile the game is and what variance we can expect from this bonus round. But that’s something for another article here in CEM.

There you have it, a quick look at expected value! Unless I happen to win the lottery, I’ll see you back here next month.

John Wilson is the Technology Editor for Casino Enterprise Management and Owner of ICS Gaming, providing slot consulting services and game design. He has designed several slot games in both Class II and Class III markets.  He can be reached at jwilson@icsgaming.com.