so the witch farm produces 6 different types of drops in equal proportions (the 7th drop, sticks, is twice as frequent as the rest, so i'm ignoring it here)

question: is it possible to predict the roughly how much the amount of drops in each category will differ?

the drops pretty much follow a poisson distribution so yes. recall that the poisson distribution is basically the same as a binomial random variable with n = lambda*k and p = 1/k as k goes to infinity but the variance of a binomial random variable is np(1-p) ~= lambda for large values of k

and in fact, lambda is a very good estimate for the variance of a poisson variable for large quantities of lambda.

the counts of my 6 drops were:

47251, 46909, 47180, 47256, 47197, and 47304, which has a sample mean of about 47183 and an unbiased sample variance of 19975 corresponding to std of about 141.33

meanwhile, the estimated variance is 47183, giving an std of 217.22

the discrepancy between theoretical variance and sample variance is not that big, but it's still big enough to make me wonder if i did anything wrong. so... we're going to compute the sample variance of the sample variance.

just kidding, we're going to substitute it with my favorite mathematical tool: simulation. assuming lambda really was 47183, what is the distribution of sample variances i'm going to get from my 6 drop types?

well i've attached it below and 141 falls pretty near the center so there's apparently no cause for concern.

tl;dr, the math checks out. so if you have about 100 of one type of mob drop, you can expect to have 100+/-10 of another mob drop which drops equally frequently. one double chest gets you almost +/- 1 stack of items. a column of 4 chests + 4 hoppers (common in a storage system) gets you +/- 2 stacks. if you 12 million of one item, it's +/- 1 double chest.

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edit: of course i just realized the far more useful application of this is that it's extraordinarily easy to put error bars on measurements of mob farm drop rates -- simply tack on sqrt(n). not sure why i didn't notice this earlier