Hypergeometric

Hypergeometric Distribution

For my article on mana base probabilities presented failure tables to help player build optimal mana bases.

A Multivariate hypergeometric distribution formulate can be thought of like this:
P = ((sA choose rA) * (sB choose rB) * (sC choose rC)) / ( N choose n)
Where:
sA = the number of successes of type A in the population
rA = the required number of successes of type A in the sample
sB = the number of successes of type B in the population
rB = the required number of successes of type B in the sample
sC = the number of failures in the sample (not A or B)
rC = the number of allowed failures in the sample
N = the population size
n = the sample size
"choose" means a binomial coefficient

In a specific case, to find the the probability of drawing exactly one of your 8 blue spells and exactly zero of your 10 blue mana sources in a hand of 7 we can calculate:
P = ((8 choose 1) * (10 choose 0) * (42 choose 6)) / ( 60 choose 7)

To get the the chance of drawing one or more spells and zero lands we have to add the probabilities of getting one, two, three, etc.

For double mana base failures we follow a similar approach, but instead of calculating only the case of zero lands we include the case with one land.

I wrote a simulation to check my tables. I spot checked a variety of results and confirmed their accuracy within a hundredth of a percent. I think my data is correct, though I'm less certain of it's meaning or that I explained it correctly here.