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title: "Approximations to the Binomial Distribution"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575260/Approximations%20to%20the%20Binomial%20Distribution"
format: markdown
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The [Binomial distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575204/) is the most fundamental distribution in probability theory. Binomial(*p,n*) models the number of successes *s* in *n* trials, where each trial is independent of others and has the same probability of success *p*. The probability of failure (1-*p*) is often written as *q* to make the equations a bit neater.
*Normal approximation to the Binomial*
When *n* is large, and *p* is neither very small nor very large, the following approximation works very well:
[Binomial](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575204/)(*p, n*) *»* [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/)(*np*, √(*npq*))
The mean and standard deviation of a Binomial distribution are *np* and √(*npq*) respectively, so this approximation is quite easy to accept. It also fits nicely with [Central Limit Theorem](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574953/), because the Binomial(*p,n*) distribution can be thought of as the sum of ***n*** independent Binomial(*p*, 1) distributions, each with mean *p* and standard deviation √(*pq*).
The difficulty lies in knowing whether, for a specific problem, the values for *n* and *p* fall within the bounds for which the Normal distribution is a good approximation. A Binomial(0.5, 1) is symmetric, so we can intuitively guess that one needs a fairly low value for *n* for the Normal approximation to be reasonable when *p* = 0.5. On the other hand, a Binomial(0.95, 1) is very highly skewed and we would reasonably expect that *n* would need to be large for the Normal approximation to work for such an extreme value of *p*. An easy way to judge this is to think about the range that a Normal distribution takes: almost all of its probability is contained within a range +/- 3 standard deviations from the mean. Now, we know that the Binomial distribution is contained within the range [0,*n*]. It would therefore be reasonable to say that the Normal distribution is a good approximation if its tails stay well within these limits, i.e.:
> Macro (mathinline)
and
> Macro (mathinline)
which simplify to:
> Macro (mathinline)
and
> Macro (mathinline)
A more stringent condition (using four instead of three standard deviations for the Normal) would be to use 4 and 16 instead of 3 and 9 in the above equations. The graph on the right shows how these two conditions work together symmetrically to show the {*p,n*} combinations which will work well. [Decker and Fitzgibbon (1991)](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26579303) advise using the Normal approximation when *n*0.31*p*>0.47, which is discussed more below. At larger values of *n*, which is when one might wish to use an approximation, their rule of thumb is somewhat more conservative than that presented here even using a range +/- 3 standard deviations from the mean.
The [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/) distribution is continuous while the Binomial distribution is discrete, and this approximation leads to an additional error that can be avoided. Rather than use *f(x)* from the [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/) distribution to give the Binomial distribution probability *p(x)*, it is more accurate to use *F(x+*0.5*) -F(x-*0.5*)* where *F(x)* is the [cumulative distribution function](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574886/) for the Normal distribution. If one is simulating the distribution with Normal, the equivalent is to use a [ROUND](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575570/)(…,0) function around the Normal distribution.
*Poisson approximation to the Binomial*
The probability mass function for the [Poisson](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/) distribution can be derived from the Binomial probability mass function by making *n* extremely large while *p* becomes very small, but within the constraint that *np* remains finite. Thus, the following approximation can be made to the Binomial:
Binomial(*p*,*n*) *»* Poisson(*np*) when *n* ® ∞ *,* *p* ® *0,* *np* remains finite
In fact the Poisson approximation works very well for relatively small values of n and large values of p. For example, a Binomial(1%,100) is very well approximated by a Poisson(1):
The [Poisson](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/) approximation tends to overestimate the tail probabilities at both ends of the distribution. [Decker and Fitzgibbon (1991)](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26579303/) advise using this approximation when *n*0.31*p*<0.47 and the Normal approximation otherwise.
We can use a [Poisson](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/) approximation to the Binomial when *p* is close to 1, i.e. as (1-*p*) *®* 0 by simply reflecting the formula. In this case, Binomial(*p*, *n*) *»* *n* - Poisson(*np*) and the Decker and Fitzgibbon condition is then *n*0.31(1-*p)*<0.47.
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