User blog:BunsenH/Monster Breeding: Uncertainties

The subject of the statistics of breeding rare monsters comes up from time to time. This is the formal derivation of the calculation of standard deviations for this kind of "experiment", somewhat adapted from my M.Sc. thesis. It's somewhat math-y, I'm afraid.

Let&#39;s start with a couple of standard equations for variance. These are right out of the textbooks and are merely how one calculates the variance of variable x in the first place, for N values in the data set. For our breeding calculations, let x be a variable which takes the value 1 if the breeding attempt is successful and 0 if it&#39;s unsuccessful. (Note also that for these values, x2 = x ; we&#39;ll need that later.) For N tries, we&#39;ll use Nm to mean the number of successful tries, and Pm is the probability of success, i.e. the fraction of tries that are successful &mdash; that is, Pm is a value between 0 and 1, inclusive. We use  x  to mean the average value of x; that&#39;s a standard notation. From these, we get:

For successful breeding attempts, x2 = x = 1, and for unsuccessful attempts, x2 = x = 0. So &Sigma; x2 = Nm. And since  x  = Pm,   x 2 =  Pm2. Plugging these into the variance equation at the top, we get:

&sigma; 2x  =    Pm&minus;  Pm2