Game Development Reference
In-Depth Information
Uniform Distribution
When a computer generates a random number, it ordinarily does so with a
uniform
distribution
. That means the chance of getting any one number exactly equals the
chance of getting any other number. It's like rolling a single die: There's an equal
chance that a die will land on any one of its faces. That is exactly the behavior you
will want whenever you ask the computer to choose among a certain number of
equally probable options. For example, if you specify that four possible answers in
a multiple-choice quiz game should be presented to the player in a random order,
you'll want the possibilities to be mixed up so that each answer has an equal
chance of being presented first, second, third, or fourth.
You can create a uniformly distributed die roll value with the following formula
(and by discarding any digits after the decimal point in the result):
Die Roll = (Random number × Number of faces on the die) + 1
Nonuniform Distribution
In other circumstances, you may not want the random values to be evenly distrib-
uted but may instead want some values to occur frequently and others to occur
only rarely. Suppose you're designing a game about Olympic archery. The player
will compete against an artificial opponent, and you want to use a random number
to decide where the artificial opponent's arrow lands. At the Olympics, the chances
that an archer will hit the bull's-eye are pretty high. The chances that he'll miss
the target entirely are extremely low. In specifying where the arrow lands, you
won't want it to be uniformly distributed across the target, you'll want it to have a
better chance of landing in the middle than anywhere else.
One of the simplest ways to achieve this result is to generate more than one uniformly
distributed random number (that is, roll several dice) and add the resulting numbers
together to give you a value. This does not yield a uniform distribution of values;
the values tend to cluster around a central point, with few values at the extremes.
Here's an example. We all know that if you roll two six-sided dice and add them
together, the chance of rolling a 7 is much higher than the chance of rolling a 2 or
a 12. But how much higher? As you design the core mechanics of your game, this is
something you need to know.
Rolling two six-sided dice and adding the results can produce any of 11 possible
values, from 2 (two 1s) to 12 (two 6s). Of 36 possible combinations of two six-sided
dice, only one combination yields a 2: throwing two 1s. On the other hand, there
are six possible ways to get a 7: 1
+
6, 2
+
5, 3
+
4, 4
+
3, 5
+
2, and 6
+
1. So when
you roll two six-sided dice, you're six times as likely to roll a 7 as you are to roll a 2.
The rules of
Dungeons & Dragons
specify that certain types of random numbers
must be generated by rolling
three
six-sided dice and adding them together. With
three dice, the chances are even higher that the result will be somewhere in the
