# Poisson Probability Question

I'm trying to solve this problem on the study guide, but I keep getting the wrong answer. The study guide gives the answer 0.677, but that doesn't help me solve the problem. Help? Thanks in advance!

The probability that a certain type of electronic component will fail during the first hour of operation is 0.005. If 400 components are tested independently, find the Poisson approximation of the probability that at most two will fail during the first hour.

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Under the given assumptions, the number $X$ of components that fail has binomial distribution, with $n=400$ and $p=0.005$. The probability that $X\le 2$ can be computed using $$\binom{400}{0}p^0(1-p)^{400}+\binom{400}{1}p^1(1-p)^{399}+\binom{400}{2}p^2(1-p)^{398}.$$
However, here $n$ is large and $np=2$ is moderate. So the binomial is well approximated by the probability that a Poisson with $\lambda=2$ is $\le 2$. This probability is $$e^{-2}+e^{-2}\frac{2^1}{1!}+e^{-2}\frac{2^2}{2!}.$$
Remark: The answer simplifies to $5e^{-2}$, which is about $0.6766764$.
I computed the "exact" answer using the binomial distribution, without the Poisson approximation. The calculator gives $0.676677$. So in this case, the Poisson approximation is very very good.
In real work, one can get away with much less precise approximations. The assumptions under which we make our calculations, such as independence, are seldom absolutely true. And the number $0.005$ is undoubtedly a rough approximation. So the numbers we obtain from a probability calculation, even if that calculation is "exact," can usually only be treated as a rough approximation to the truth.