Probabilities Function -

This is a bit of a math question, and cause am quite weak at math [ :( ] I can't figure this out

I have an application that must "randomly" decide if you won or not with a maximum daily winners, the problem is that i don't want to do a simple x chance of winning cause this might result in 20 people winning at the start of the day, and then everyone will keep losing, is there a generic formula to do this?

tl;dr

I have x amount of Gifts (x=20) And I want to randomly spread them throughout the day, is there a generic function/script?

Any ideas?

Edit: For the first day at least, there is no predictable number of daily contestants