I think I can solve this problem using hypergeometric distributions, but the solution is ugly and complicated (it gives 31.8 draws on average are needed, which matches Monte Carlo simulations a colleague ran for me). Note that I am stopping after pulling the third jack, so my last draw must be a successful jack draw. On average, how many cards did I need to pull before reaching 3 jacks? I repeat this infinite times, and then average the number of cards pulled to reach 3 jacks. in the first experiment I might have hit the 3rd jack on the 40th card, so I write down '40'. Jacks, I write down the number of cards I had needed to pull and stop the experiment. In each experiment, I shuffle the deck and pull cards one-by-one and discard the card without replacement. Let's say I have a goal of drawing 3 jacks from a regular deck of 52 cards (in which there are 4 jacks). I have the following probability problem that I think must be quite common.
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