Expected value vs probability

Question

Sorry in advance, it is probably a stupid question. I encountered it when I was thinking about the birthday problem. The probability of having at least one pair of the same birthday is $$ 1- \frac{365\cdot364\cdot\ldots\cdot(365-n+1)}{365^n}$$ and it is above 0.5 for n>22. However the expected value for the number of pairs is $$ E[\# \text{pairs of same birthdays}] = \binom{n}{2}\cdot\frac{1}{365}$$ which is larger than 1 for n>27.

So why is there a difference?

Answer 1

The classical birthday paradox computation concerns itself with the median of the number of pairs of same birthdays, which is not the mean or expected value. The two are usually not the same.

Answer 2

There may be more than one match, which increases the expected value.