Binomial/Poisson mean value formula

A year or so ago, I was looking into a problem that involved a lot of coins with different probabilities of flipping heads. The probability that $\displaystyle n$ heads would be flipped among all of the coins follows a Poisson distribution. I saved an article that I planned to look over more carefully that discussed using the mean probability of flipping heads along with the standard deviation among the coins to convert it to a system that seemed (to me) more similar to a Binomial distribution. I seem to recall that it involved a formula that used the cube of the standard deviation (maybe), but beyond that, I can't recall much about the article. I have no clue where I saved it. Does anyone know where I can find the article again? I apologize for my lack of information. I really have no clue what I did with it.

Re: Binomial/Poisson mean value formula

Hey SlipEternal.

What exactly are you trying to do?

Re: Binomial/Poisson mean value formula

I am trying to create an AI for a board game with rules very similar to Axis and Allies. To develop the AI, I wanted to understand the battle statistics better. Unfortunately, battle statistics are rather difficult to obtain. The most accurate results come from direct calculations of probabilities of results, but estimates from running simulations can give me a good enough idea. Unfortunately, the simulations are more computationally complex than I'd like, so I want a method that is computationally more efficient than either of these methods (allowing me to run the results of many different types of battles instead of just a few).

Essentially, different units can be considered "coins" that flip heads with different probabilities. So, an army can be thought of as "pools" of coins. The "pools" would be coins that flip heads with the following probabilities: 1/6, 1/3, 1/2, 2/3, 5/6, and 8/9. For a single round of combat, both players "flip all their coins" and count the number of heads they get. Their opponent then removes that number of coins (of their choice).

So, let A and B be vectors where the dimensions are the number of units of each type. Let P(A,B,n) be the probability that in a battle between the units in vectors A and B, A wins and sustains exactly n casualties. For a vector v of units, let h(v) be the vector of units after v "takes a hit". Let d(A) be the vector of numbers of coins in each coin pool associated with the units of vector A. Then, let P(d(A),k) be the probability that the units in vector A score k hits against its opponent in a single round. P(d(A),k) has a binomial distribution if d(A) consists of only one pool of coins. If d(A) consists of multiple pools of coins, P(d(A),k) can be approximated by a Poisson distribution. I can find the exact distribution, but I need to calculate potentially n! different terms to determine the exact probabilities I am looking for (where n is the number of "coins" in all pools). This is even more computationally complex than running simulations. So, I want to come up with a method that will get me closer than a Poisson distribution, but with less computational complexity than the actual solution.

Then, I can use those results to calculate P(A,B,n) where

$\displaystyle P(A,B,n) = \sum_{i=0}^{|A|}\sum_{j=0}^{|B|}P(d(A),i)\cdot P(d(B),j)\cdot P(h^j(A),h^i(B),n-j)$

and $\displaystyle \forall n\neq 0, P(0,B,n) = P(A,0,n)=0, P(A,0,0)=1$