I understand that if there are 253 comparisons to be made when checking if any two people out of 23 share the same birthday, that it's 1 minus the probability of them not sharing a birthday (.99726) to the power of 253, and that makes perfect sense, like flipping a coin multiple times and multiplying the probabilities of each trial together. $$P = 1 - (.99726)^{253}$$
But I can't figure out why just simply putting 253/366 gives you the wrong answer. If there are 253 comparisons and 366 days why does this not work. I know it's wrong I just really want to understand why, what am I missing?
Thanks in advance, Richard
It turns out that $\frac{253}{366}$ is the expected number of pairs of people with the same birthday (assuming each day is equally likely.) But in some sets of 23 people, you'd have more than one such pair. So the expected number of pairs will be strictly greater than the probability that a pair exists.
Essentially if $X$ is the number of pairs of people with the same birthday, and $$Y=\begin{cases}1&X\geq 1\\ 0&X=0\end{cases},$$ then $E(X)=\frac{253}{366}$ and $E(Y)=P(X>0)$ is the probability some of the $23$ people have the same birthday. We know $Y\leq X$ with strict inequality sometimes, you have $E(Y)<E(X).$
But your first answer is wrong, too. The probability that they don't have the same birthday is:
$$\frac{366}{366}\cdot\frac{365}{366}\cdots\frac{366-22}{366}=\prod_{k=0}^{22}\left(1-\frac{k}{366}\right)$$
This is not equal to $\left(1-\frac{1}{366}\right)^{253}$, although it is not a terrible estimate since $1-\frac{k}{366}\approx \left(1-\frac{1}{366}\right)^k$ for $k$ small.
Bernoulli's inequality gives us $1-\frac{k}{366}<\left(1-\frac{1}{366}\right)^k,$ so we have that $P>1-\left(1-\frac{1}{366}\right)^{253}$.