medium Probability
A release-checked medium problem for training Expected Value, Discrete Random Variables.
Suppose you have $20$ cards with the values $1-10$ each appearing twice in a deck. You draw $2$ cards at a time uniformly at random from the $20$ cards. If they match in value, you remove them from the deck. Otherwise, they are put back into the deck. The game finishes once there are no more cards to draw. Each drawing of two cards is a turn. Find the expected number of turns needed to finish the game.
This Probability problem is tagged Expected Value, Discrete Random Variables. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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