# 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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