Thursday, February 25, 2010

Average value of a fair die.

Just for completeness, we will show why the average value of a fair dX die is (X+1)/2.

A fair dX die will have X sides with the numbers 1,2, ... to X, which have an equal likelihood of being rolled. Hence the probability of rolling any number is 1/X.

The average value is the sum of the die side values weighted by the probabilities of being rolled. Mathematically, for example the average value for d4 is:

1(1/4) + 2(1/4) + 3(1/4) + 4(1/4) = (1+2+3+4)/4

while for a d6, the average value is:

1(1/6) + 2(1/6) + 3(/16) + 4(1/6) + 5(1/6) + 6(1/6) = (1+2+3+4+5+6)/6

In general, the average value for a fair dX die is:

1(1/X) + 2(1/X) + ... + (X-1)/X + X/X = [1+2+...+(X-1)+X]/X

Recall from high school algebra, the sum of the first n numbers 1, 2, ..., n-1, n is: n(n+1)/2. (This can be shown to be true for any n by mathematical induction).

So for the average value of dX, we get: [X(X+1)/2]/X = (X+1)/2