Advent

Elementary Probability Theory 3

by Paul Mitchener
Oct 26,2005

Methods of Campaign and Adventure Design 13

By Paul Mitchener

Elementary Probability Theory 3

It is quite common to hear people talking about bell curves, and saying such things as '3d6 is a bell curve whereas d20 is not'. As we saw in the last column, 3d6 is less random than d20 (it has a lower standard deviation). To be pedantic, however, 3d6 does not have a bell curve distribution, although it is quite close to such a distribution.

Random variables with bell curve distributions are called normal or Gaussian random variables. In this column I want to talk about such random variables and why they are useful. To do this properly, some calculus is unfortunately needed. However, I hope to be able to keep discussion of calculus to an absolute minimum and to keep the column coherent even for a reader with no knowledge of calculus.

Continuous Random Variables

Let X be a discrete random variable. Then for every real number a, we can define the probability that X is less than or equal to a. Let us write this probability F(a). The function F(a) is called the distribution function of the random variable X. It has the following three properties:

• If a is less than b, then F(a) is less than or equal to F(b).
• As the number a gets higher and higher, the value F(a) gets closer and closer to 1 (mathematically speaking, the limit of F(a) as a tends to infinity is equal to 1).
• As the number a gets lower and lower, the value F(a) gets closer and closer to 0.

We can now generalise the definition of a discrete random variable. A (not necessarily discrete) random variable X is a function that assigns a probability F(a) to each real number a. The value F(a) is the probability that X is less than or equal to a. The function F must satisfy the three properties above and is called the distribution function of X.

Now for some calculus. A random variable X is called continuous if there is a function f(x), called the probability density function such that the distribution F(a) is equal to the integral of the function f(x) on the set of all points less than or equal to a.

The value f(x) of a probability density function measures the probability that the random variable is close to the number x. Thus, the higher the value f(x), the greater the probability that a random variable is close to x.

The expectation and standard deviation of a continuous random variable can be defined as integrals. I will not give the precise definitions here. As usual, the expectation is the average value of a random variable, and the standard deviation is a measure of the average distance a random variable is from its expectation.

Normal Random Variables

A normal random variable is a continuous random variable with probability density function of the form.

• f(x) = (1/sqrt(2(pi)a^2) exp (-(x-m)^2/2a^2)

The number m is the expectation of the random variable, and the number a is the standard deviation. The graph of the function f(x) is a bell curve. The centre of the bell curve is the mean m. The higher the standard deviation a, the more spread out the curve is.

The 67%/95% rule applies exactly to normal random variables. This rule says that for a normal random variable, 67% of all possible values are no further from the expectation than the standard deviation. 95% of all possible values are no further from the expectation than twice the standard deviation.

The Central Limit Theorem

Let X1,X2,X3,... be random variables, each with expectation m and standard deviation a. Then the central limit theorem tells us that the sequence of variables ((X1 + ...+ XN)-mN)/sqrt{N} gets closer and closer to a normal random variable with expectation 0 and standard deviation a.

The central limit theorem means that normal random variables are fundamental, and many random variables one comes across in real life where the distribution is unknown can be approximated by normal random variables. In RPGs, this means that randomising methods involving normal random variables often 'feel right' in terms of how randomly things turn out.

Of course, normal random variables are continuous rather than discrete, and therefore cannot be generated exactly by rolling dice. However, if we have N dice, each with X sides, then the random variable NdX is a sum of independent random variables. For large N, this random variable will be close to a normal distribution.

Actually, such a random variable can be quite close to a normal random variable for relatively low values of N (around 3 or more). This is what is meant when people say (for example) that '3d6 is a bell curve'. Actually, it is not a bell curve, but is a reasonable approximation. And the philosophy I mentioned above might give some indication as to why this is thought to be a good thing.

Wrapping Up

Okay- this was the last of my promised series of three probability columns. Next month there are two possible ideas. The first idea is to go back to talking about campaign design by looking at campaign prospectuses. Alternatively, if someone suggests some dice mechanic in the feedback, I could try to analyse it. As a warning, I am not too familiar with many roll and keep systems, so an exact explanation will be needed before I can do any analysis.