I have the following probability problem that I think must be quite common. The problem is as follows:

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> 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 conduct many experiments. In each experiment, I shuffle the deck and pull cards one-by-one and discard the card without replacement. When I reach my goal of having pulled 3
>  jacks, I write down the number of cards I had needed to pull *and stop the experiment*. e.g. in the first experiment I might have hit the 3rd jack on the 40th card, so I write down '40'. 
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> 
> I repeat this infinite times, and then average the number of cards pulled to reach 3 jacks. On average, how many cards did I need to pull before reaching 3 jacks?
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Note that I am stopping after pulling the third jack, so my last draw must be a successful jack draw.

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). I think I've stumbled upon a much simpler formula:

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> average draws needed = (n) \* (x+1)/(y+1)
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where n is the number of jacks I want (3), x is the number of cards in the deck (52), and y is the number of jacks in the deck (4). 

Other than by blind luck of simple observation that I got playing around with numbers, I have no idea how to derive the above formula. 

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> Has anyone seen this problem and know how this simple formula might be derived?
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I should also note that the simple formula has been tested for many n, x, and y values and matches both the complicated formula and several simulations run for this problem. So there is a decent degree of confidence that it is correct.