My question is related to this question.
I am wondering if we have a large crowd of $2000$ people: what is the expected minimum number days of the year we have to pick in order for at least half of the birthdays of the crowd to be in one of those days?
To be clear: with $2000$ people the average number of birthdays for any day is between $5$ and $6$, but there surely will be days where $10$ or even $12$ people have their birthday. So, knowing the birthdays of all people, I want to pick as many of those days, so that with as few days as possible I do cover half of the birthdays. That should be a good bit below $183$ I would think.
In fact, my intuition is that for a crowd of $2000$, this would be $100$ or so, but I could be way off, so I would like to know.
I also have the feeling that with a crowd of this size, we can make a fairly precise prediction as to how many days are needed. That is, if the expected number of days we need to cover at least $1000$ of the $2000$ birthdays is $100$, I would guess that the actual number of days with an actual crowd will be fairly close that that expected number. That is, I would like to have a result that says something like: There is a $95$% chance that the actual number of days lies within the interval $[X-Y,X+Y]$.
I am just guessing: $X=100$ and $Y=5$
How far am I off?
I welcome any kind of mathematical analysis, but I also welcome computer simulated approximations for this. And of course, while I am particularly interested in a crowd size of $2000$, feel free to provide a general answer in terms of $n$.