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This question already has an answer here:

There're k balls in a bag, each one with a different colour. Draw one ball from the bag randomly each time, then put the ball back.

Let random variable X = the number of balls to draw until all k colours have been seen. Find E(X).

Please help me with this question, thanks!

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marked as duplicate by joriki probability Jun 18 '16 at 1:07

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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This is the "coupon collector's problem". As the Wikipdia article explains, the expected value is $k$ times the $k$th harmonic number: $$ k\left( 1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 k \right). $$

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