### Iskandria

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#### Mass event resolution

Often, for game purposes, a number of events (attacks, etc.) with equal probability of success can be resolved consecutively. Furthermore, details about the success can be deferred. If suitable electronic support is available (spreadsheets or scientific calculator with the combinatorial function C(n,k), where k,n are non-negative integers such that 0<k<n), the following procedure for resolving how many successes are in a mass event is to be used:
• There is a probability for success P(success), and a corresponding probability of failure P(failure), related by P(success)+P(failure)=1. The larger of these is denoted by the variable q, and the smaller of these is denoted by the variable p. [If both are 1/2, obviously the precise allocation doesn't matter.]
• The calculational procedure assumes that both P(success) and P(failure) are in a form suitable for entering for electronic calculation...that is, decimal approximations. If both P(success) and P(failure) have explicit analytic formulae, explicitly calculate the smaller of the two [p] and use the transformed constraint q=1-p to get q for the calculator.
• Because we are preparing numerical approximations for further calculation, we have to consider the truncation error that results from subtracting two numbers...which manifests as a loss of significant digits (which impairs calculations derived from the resulting intermediate data). EXERCISE: Explain why the stated procedure generally minimizes truncation error for both p and q compared to the other way around.
Now, we are going to use the Binomial Theorem:
(p+q)n = Σk=0..n C(n,k)pkqn-k
Each term has a significance:
P(exactly k events of probability p)=C(n,k)pkqn-k
Observe that there is a minimal m such that a pseudorandomly generated (decimal) number R from the uniform distribution over the interval [0,1] (a number between 0 and 1) satisfies
R < Σk=0..m C(n,k)pkqn-k
This m is the number of events with probability p in the mass event. n-m is the number of events with probability q in the mass event. Detail resolution depends on what the mass event is.

The attentive reader will notice that this does not necessarily precisely compute boundaries near 1...which does not seem to be a significant problem, in practice. More problematic is that the direct computation of m (by adding terms until it is found) is tedious for large n as p approaches 1/2 from above.

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