Poisson Approximation of Binomial Probabilities

Poisson Approximation of Binomial Probabilities

The logic and computational details of binomial probabilities are
descriped in Chapters 5 and 6 of Concepts and Applications.

This page need be used only for those binomial situations in which n is very large and p is very small. For example: The null hypothesis holds that a certain genetic characteristic will express itself in p=.001 of the population. In a sample of n=3000 subjects, k=7 are observed to display the characteristic, whereas only np=3 are expected. On the null hypothesis, how likely is it that a rate this great or greater could occur by mere chance? Your computer would not be able to perform the factorial and exponential operations required for direct calculation (Exact Binomial Probability Calculator), and np<5 would preclude use the normal approximation (Binomial z-Ratio Calculator).

In cases of this sort, the appropriate binomial probabilities can be approximated by way of the Poisson probability function


	^TP_{(k out of n)} =	(e^—np)(np^k) k!

where:

	e =	the base of the natural logarithms;
	n =	the number of opportunities for event x to occur;
	k =	the number of times that event x occurs or is stipulated to occur; and
	p =	the probability that event x will occur on any particular occasion;

Application of the Poisson function using these particular values of n, k, and p, will give the probability of getting exactly 7 instances in 3000 subjects. Applying it to all values of k equal to or greater than 7 will yield the probability of getting 7 or more instances in 3000 subjects, while applying it to all values of k equal to or smaller than 7 will give the probability of getting 7 or fewer instances in 3000 subjects. Note, however, that these results are only approximations of the true binomial probabilities, valid only in the degree that the binomial variance is a close approximation of the binomial mean.

To perform calculations of this type, enter the appropriate values for n, k, and p (the value of q=1—p will be calculated and entered automatically). Then click the 'Calculate' button. To enter a new set of values for n, k, and p, click the 'Reset' button. The value entered for p can be either a decimal fraction such as .001 or a common fraction such as 1/1000. Whenever possible, it is better to enter the common fraction rather than a rounded decimal fraction: e.g., 1/1050 rather than .00095.

Poisson Approximation
of Binomial Probabilities

n	k	p	q

binomial mean
binomial SD
discrepancy

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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 mean and variance of a binomial sampling distribution are equal to np and npq, respectively (with q=1—p). As p approaches zero, the value of npq approaches that of np, and the binomial distribution accordingly approximates the form and properties of the Poisson.

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