I have a doubt regarding when to approximate binomial distribution with Poisson distribution and when to do the same with Normal distribution.

It is my understanding that, when p is close to 0.5, that is binomial is fairly symmetric, then Normal approximation gives a good answer. However, when p is very small (close to 0) or very large (close to 1), then the Poisson distribution best approximates the Binomial distribution.

Also, when n is large enough to compensate, normal will work as a good approximation even when n is not close to 0.5 (n will work fine, but still Poisson will be better? )

However,consider the following question-

`The probability of any given policy in a portfolio of term assurance policies lapsing before it expires is considered to be 0.15. For a group of 100 such policies, calculate the approximate probability that more than 20 will lapse before they expire. `

Here n is 100 and p is 0.15 (which is not close to 0.5). In this case, the **exact answer is 0.0663**. The **normal approximated answer is 0.06178** and the **Poisson approximated answer is 0.08297**.

My doubt is that, **since p is closer to zero than it is to 0.5, shouldn't the Poisson approximation yield a better answer?**

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#### Best Answer

Here is a pmf plot I was able to create in MATLAB—looks like the normal (Gaussian) is pretty close, where as the Poisson misses the peak and has a fatter long tail.

Furthermore, looking at wiki (not always infallible!), according to NIST/SEMATECH, "6.3.3.1. Counts Control Charts", e-Handbook of Statistical Methods., Poisson is a good approximation for $p < 0.05$ (not 0.5) for $n > 20$.