# Why can we use poisson distribution in this case? the n is very small!!

Suppose 8% of the tires manufactured at a particular plant are defective. To illustrate the use of the Poisson approximation for the binomial, the probability of obtaining exactly one defective tire from a sample of 20 is calculated as follows:

in this case, the n is only 20!

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It is a approximation at the end of the day - the natural question is for what values of $n$ and $p$ is the approximation reasonable. You might initially think that $n$ should be huge for this to work, but as the examples in http://courses.wcupa.edu/rbove/Berenson/10th%20ed%20CD-ROM%20topics/section5_6.pdf show, it can work even for $n=20$ and $p=0.08$.

If you look at how the poisson distribution is obtained from the binomial, as derived in http://www.stat.yale.edu/~pollard/Courses/241.fall97/Poisson.pdf, you see there are two main approximations.

First,

$$\frac{n(n-1)\ldots(n-k+1)}{n^k} \approx 1$$

The above is not a bad approximation for $n=20$ and $k=1$. As you increased k you would see begin to see a problem soon. It is also dependent on how much accuracy you need.

Secondly, you also approximate $(1-\lambda/n)^{n-k} \approx (1-\lambda/n)^{n} \approx e^{-\lambda}$ which is a pretty good approximation for moderately large $n$ values and small $k$.

Since these are the two approximations involved, we can be confident about using the poisson distribution whenever they are satisfactory, which in this case is true even though $n$ is only 20.

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are you sure $$$\frac{n(n-1)\ldots(n-k+1)}{n^n} \approx 1$$$? because (by looking at how it's derived) the denominator is $k!$ –  user133466 Oct 7 '12 at 0:56
The denominator should be $n^k$ and I've edited it accordingly. The $k!$ is part of the final poisson distribution expression. I brought the $n^k$ in from the $(\lambda/n)^k$ term. –  svenkatr Oct 7 '12 at 1:15