Show that $p_{n} \geq 1- \exp{(-n(n-1)/730}$

by MinaThuma   Last Updated February 11, 2019 10:20 AM

On the issue of the birthday paradox,Let $p_{n}$ be the probability that in a class of $n$ at least $2$ have a their birthdays on the same day (exclude $29$ Feb). Use the inequality $1-x \leq e^{-x}$ to show that:

$p_{n} \geq 1- \exp{(-n(n-1)/730}$ and then determine $n \in \mathbb N$ so that $p_{n} \geq \frac{1}{2}$

My ideas:

First $p_{n}=1-\frac{\frac{365!}{(365-n)!}}{365^{n}}$ using the inequality given to us.

$1-\exp{(-\frac{\frac{365!}{(365-n)!}}{365^{n}})}\geq p_{n}$, what am I supposed to do next? Use Stirling's Formula?



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