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A combinatorial problem
From a full deck of 52 cards, we draw one card at random. We look at the card and put it back into the deck. Then we shuffle the deck and again draw one card at random, look at the card, and put it back into the deck. We repeat these steps as many times as necessary, until we have seen every card in the deck. How many times, on the average, do we need to draw a card?
My answer is 235.48 times. I don't know how to solve the problem easily or elegantly. I produced a solution that is somewhat long and cumbersome; I computed a generating function for the distribution of the required number of draws. But maybe that's just me; I don't have a good intuition for combinatorical problems, and I usually start by deriving generating functions for everything. |