What's the probability of their being at least one student born on each day of the year (365) in a class of 2500?

I think:

2500               
_____       :    (365*364*363*362...4*3*2)

365

(500/73):365!

It's almost 5:30am, I'm sleepy, so I wrote this up quickly, and it probably has a mistake somewhere, but at this hour I don't care

First, the probability that on some particular day (say Jan 1st) no students had their
birthday on it is : (probability 1st student wasn't born on Jan 1st) * (probability 2nd
student wasn't born on Jan 1st) * ... * (probability 2500th student wasn't born on Jan 1st)
= (364/365) ^ 2500 = 0.0010502

Now, if we add up

(probability no students had their birthday on Jan 1st) +
(probability no students had their birthday on Jan 2nd) +
(probability no students had their birthday on Jan 3rd) +
...
(probability no students had their birthday on Dec 31st)

= 0.0010502 * 365 = .383323

we will get the probability that there is at least one day in the year where no students had
their birthday. 1 minus that, is the probability that there is no such day, i.e. that on
every day someone has their birthday, which is what we want. So that will be 1 - .383323 =
.616677.

What about the fact that a solar year is closer to 365.2425 days long, and there are people born on Feb. 29th?

Is it safe to assume that one is equally likely to be born on any day of the year? Are there any dates that births are less likely to happen (say, holidays)?

I don't think births are less likely to happen on any given day. But I'd assume that conception is less or more likely to happen on certain days.

Well there are fewer births on weekends than weekdays, and that's a statistical fact. So the same might apply to holidays as well. (Think inductions, planned c-sections etc.)

Yeah, I thought about inductions and planned c-sections. but I still don't think it's such a high number that it should make a big difference (maybe I'm too naive?). Thinking about the weekends, it's weird, but when I was pregnant and had contractions for weeks, the cx picked up every single weekend even though I didn't change my level of activity at all (I was on bedrest).

Thanks.

Define "safe." Probabilities are always models, not reality. Int's assumption was that birthdays are picked at random from a regular distribution. If there was a peak around a particular set of days, one could change the distribution and therefore the results. In the absence of knowledge of such a peak, his assumption is the best that can be done.