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Suppose we have shuffled a deck with $n$ red cards and $n$ black cards. What is the expected number of cards we must draw before we see our first red card? Before we see all the red cards?

Which distribution should be used? I can work out some numerical examples but they do not point to anything general.

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This is similar (in technique) to this question that came up yesterday. In particular the various methods discussed there apply here.

I'll write out the solution for a standard deck of cards, $n=26$, though the method applies generally.

To take the simplest method, see the solution of @paw88789 : Think of the $26$ Red cards as separators for runs of Black cards. As there are $26$ separators, there are $27$ runs. It follows that the average length of a run is $\frac {26}{27}$ Whence the answer to the first question is $$1+\frac {26}{27}=\frac {53}{27}$$ and the answer to the second is $$52-\frac {26}{27}=51.\overline {037}$$

Sanity check: were we to draw with replacement then the answer to the first part would be $2$, making $1.96$ quite plausible.

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  • $\begingroup$ excellent solution $\endgroup$ – Andre Mar 2 '16 at 1:23

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