I'm currently doing an exploration of the Birthday Problem, and noticed that the formula given to calculate the probability for $m$ people in a room is:
$$1-\frac{365!}{365^m (365-m)!}$$
And this can be approximated by the Poisson approximation:
$$1-e^{-\lambda}$$
Where $$\lambda = np = \frac{m(m-1)}{730},$$ for $m$ number of people in a room.
I have calculated the probabilities for $m = 1$ to $m = 365$ and they form a scatter plot.
My question is, how could one go about calculating the accuracy of the Poisson Approximation to this probability?
So far I looked into the Z-test for two proportions, but I'm not sure if that is an appropriate test or should I be plotting two scatter plots, one for each formula?
The idea here is that I want to identify if the Poisson function is a good way to approximate this probability.
Thank you in advance.
I'm not sure I know exactly what you mean by calculating the accuracy. If you want to know the error you made when calculating $f(m)$ with the function $\tilde f(m)$, simply calculate $$|f(m)-\tilde f(m)|$$ which gives you the absolute error of your approxumation. The relative error is $$\frac{|f(m)-\tilde f(m)|}{|f(m)|}.$$