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Dice Probability

Another book that explores probability in games in a completely different manner is Dice Games Properly Explained, by game designer Reiner Knizia. In addition to providing an extensive library of dice games, the book contains a chapter that offers a non-technical model of dice mathematics as a foundation for understanding probability. Depending on your experience with and love for mathematics, the following sections might seem ridiculously facile or excruciatingly dry. But it is important in this schema to establish a few basic concepts for understanding what probability in games is all about.

As the first example of probability, consider a single, standard die which has six sides, each side displaying a number from one to six. Knizia calls the numbers one to six the basic outcomes of throwing a die. Each time you throw a die, each basic outcome has a 1/6 or 16.67 percent chance of appearing. If you add up all of the chances of basic outcomes, they total to 1 or 100 percent.

Knizia lists three qualities of any basic outcome on a die:

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  All basic outcomes are equally likely.



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  The process always produces one of the basic outcomes.



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  The probabilities of all basic outcomes add up to 1.[3]
  

A combined outcome is a result that puts together more than one basic outcome.To determine a combined outcome, add the basic outcomes. For example, rolling an even number on a single die means rolling a 2, 4, or 6. These three basic outcomes add up to 3/6, so the combined outcome of rolling an even number is 50 percent. Knizia calls rolling a combined outcome an "event" and concludes that, "the probability of any event is calculated by adding up the number of the desired basic outcomes and dividing by the number of all possible basic outcomes." [4]

If you roll more than one die, determining outcomes becomes more complex. With two dice, Knizia notates the basic outcomes in the form of 3~4, where the first number is the number rolled on the first die and the second number is the number rolled on the second die. A chart of all of the possibilities of basic outcomes for two dice are shown in Table 1.

Note that symmetrical outcomes such as 2~5 and 5~2 both appear on the chart, as they represent different possible basic outcomes. There are 36 basic outcomes with two dice, so the chance of any one outcome appearing is 1/36. When two dice are thrown in a game, instead of the individual results on each die, the game uses the combined total of both dice. We can determine the chance of rolling a combined outcome equal to a particular number in the same way as with a single die: by adding up the basic outcomes. To determine the chance of rolling a 5, count the basic outcomes that add up to 5: 1~4, 2~3, 3~2, 4~1. This is four basic outcomes, and 4/36 = 1/9 or 11.11 percent.

Table 1

What is the chance of rolling doubles? There are six basic outcomes that are doubles: 1~1, 2~2, 3~3, 4~4, 5~5, 6~6. This is six outcomes, and 6/36 = 1/6 or 16.67 percent.

Putting all of the two-die outcomes on one chart, we get the figures in Table 2. Notice the radically unequal distribution of probabilities for rolling the highest and lowest numbers, as opposed to rolling numbers in the middle of the range.

Table 2

We can apply the same basic principles Knizia articulates to a wide variety of game design situations. For example, if your game requires players to flip a coin, you can determine the two basic outcomes as heads and tails, with a 50 percent chance of achieving each outcome. Or you might be designing a computer simulation game that uses lots of random numbers to determine the frequency of events, or a special deck of cards with a particular chance of certain cards appearing each turn. In any of these cases, the general principles remain the same. If players are flipping three coins, how likely is it that they will all come up heads? What is the chance of having a hand of five cards that are all the same?

If you are designing a game that involves dice-rolling, card-shuffling, or other forms of random number generation, it is important that you understand the basic principles of the probabilities involved. However, mathematical principles alone won't lead you to design meaningful play. The key, as with other aspects of games, is in understanding how probability relates to player decisions and outcomes. For example, in designing a board game such as Monopoly, in which players' pieces circle the board on a track, how will you determine the number of spaces on the board? In Monopoly, the board has forty spaces. Because the average combined outcome of a two-die roll is 7, it takes on average six throws to get around the board. This means that by about turn seven, some of the players will likely have already started their second loop, and will begin to land on each other's properties. If you are creating a game with a similar structure, design the board and the use of dice to achieve a pacing of events appropriate for your game.

[3]Reiner Knizia,

Dice Games Properly Explained (Tadworth, Surrey: Right Way Books, 1992), p. 62.

[4]Ibid. p. 63.