Let suppose that I have a box with $k$ different balls, each one with a different color.


At each time I have to extract a ball and observe the color. Then I put the ball back in the box.


How many extraction I need in order to see all the colors with probability at least $1-\alpha$?


This looks like a multinomial distribution, but instead of finding the probability of a certain result, I have to find the number of trials.



Multinomial -- how many trials in order to see all the values with prob 1-\alpha
This is the Coupon collector's problem. Check out <http://en.wikipedia.org/wiki/Coupon_collectors_problem>, in particular, the section on tail estimates.


