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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.

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This is the Coupon collector's problem. Check out http://en.wikipedia.org/wiki/Coupon_collectors_problem, in particular, the section on tail estimates.

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