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Suppose I have a fair, six-sided die. What is the expected number of rolls it would take me to get four distinct outcomes?

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Use Geometric probability, this is called Coupon Collector's problem.

Mean number of rolls until the first: $\mathbf{E}X_1 = 1$

Mean number of rolls until the second: $\mathbf{E}X_2 = \frac{1}{\frac{5}{6}} = \frac{6}{5}$

Can you handle from here?

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    $\begingroup$ What about if I wanted to do 75% of the faces on an N sided dice? How would I do that? I'm assuming the answer is a function $\endgroup$ – ToedStorm May 19 at 10:40

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