A text file contains 6000 characters. When the file is sent by e-mail from one machine to another, each character (independently of all other characters) has probability 0.001 of being corrupted. Use a Poisson random variable to estimate the probability that the file is transferred without error.
Compare this to the answer obtained when you model the number of errors as a binomial random variable.
For the binomial probability I got 0.2471%(to 4 significant figs).
For the Poisson probability I got 0.2478%(to 4 significant figs).
However I'm not sure how I'm supposed to compare them, clearly I can see that the binomial probability is slightly lower, but I don't understand why this is the case?
Let $n=6000$, $p=\frac1{1000}$, and $X\sim\mathrm{pois}(n,p)$. Then $$ \mathbb P(X=0) = e^{-\frac np} = e^{-6} \approx 0.002478752. $$ Now let $Y\sim\mathrm{Binom}(n,p)$. Then $$ \mathbb P(Y=0) = (1-p)^n \approx 0.002471322. $$ Since the difference is $-7.43006\times10^{-6}$, this is negligible.