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Suppose I have a standard deck of cards (26 red cards and 26 black cards) shuffled into a random order. I draw the cards one at a time (not replacing them) until I go through the entire deck. What is the probability that I make it through the entire deck with the number of red cards drawn never exceeding the number of black cards drawn? (More general solutions obviously welcome and likely to eventuate anyway).

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What you are looking for is the number of Dyck words of length $52$, divided by the total number of distributions of red and black cards, given by $\binom{52}{26}$. The number of Dyck words of a given length is counted by the Catalan numbers, and the final answer is $\frac{1}{27}$.

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  • $\begingroup$ Thank you very much! I hadn't come across Dyck words before - very interesting! $\endgroup$ – John Sep 26 '17 at 13:01

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