Method 1: If n
1000, exact binomial probabilities will be calculated through repeated applications of the standard binomial formula
Q
| P(k out of n) =
| n! k!(n-k)!
| (pk)(qn-k)
|
In principle,
Method 1 is preferable in all cases, since it involves direct calculation of exact binomial probabilities. Its limitation is that it is not computationally feasible with very large samples. The programming on this page is capable of performing the calculation up through
n=1000.
Method 2: If np5 and nq5, binomial probabilities will be estimated by way of the binomial approximation of the normal distribution, according to the formulaQ
| z =
| (kM )±.5 -
|
where:
-M =
| np [the mean of the binomial sampling distribution]
- =
| sqrt[npq] [the standard deviation of the binomial sampling distribution]
| |
Method 3: If n≥150 and the mean (np) and variance (npq) of the binomial sampling distribution are within 10% of each other, binomial probabilities will be estimated through repeated applications of the Poisson probability function
|
| TP(k out of n) =
| (eM)(Mk) k!
|
where:
| e =
| the base of the natural logarithms; and
M =
| np [the mean of the binomial sampling distribution]
| |
The defining characteristic of a Poisson distribution is that its mean and variance are identical. In a binomial sampling distribution, this condition is approximated as p becomes very small, providing that n is relatively large. The programming on this page permits the Poisson procedure to be performed whenever np and npq are within 10% of each other, providing that n≥150. Do keep in mind, however, that the results of the Poisson procedure are only approximations of the true binomial probabilities, valid only in the degree that the binomial mean and variance are very close.
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