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A Question About Dice

I was reading " Expected time to roll all 1 through 6 on a die " and it got me thinking...

There are various ways to generalize the coupon-collector's problem (for example, by requiring certain tickets to be collected a certain number of times minimally... I think you can abuse exponential generating functions for that one).

The generalization I had in mind is this: The coupons have an arbitrary distribution: e.g. some are rarer than others.

Thus, given a discrete distribution $p_{coupon=x}(x)$, is there a solution that is more elegant than doing a massive calculation over the entire tree of outcomes with factorially-many branches? Also does the solution generalize nicely to continuous distributions, and if so, what does it mean to have the "coupon collection" problem on a continuous distribution?

A simple case might be, for example, how many rolls it takes to see all outcomes (2-12) of rolling a pair of dice. Of course there's a nice solution, but if I were to give an arbitrary distribution...

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    Yeah, it's an interesting problem. Many variations of the original problem have been investigated. I would use Google and start looking around. Have fun!2011-06-08

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