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Player 1 continuously draws one card each time (without replacement) from a deck of cards and stops when Player 1 gets 3 of Hearts.

Player 1 gives the first card he draws to Player 2. From then on, whenever Player 1 draws a card that is smaller than the value of the previous card that he gives to Player 2, he gives the new card to Player 2 (even if the card is 3 of Hearts).

What is the expected number of cards that Player 1 gives to Player 2?

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A card is given to Player $2$ exactly if it comes before all cards smaller or equal to it and before the $3$ of hearts. In a standard deck, there are $13$ different values and $4$ suits of each value, and a card with the $k$-th smallest value has $4k-1$ cards smaller or equal to it. By symmetry, the probability that the card comes first out of the $4k-1+1+1=4k+1$ cards ($4k-1$ smaller or equal cards, the $3$ of hearts, and the card itself) is $\frac1{4k+1}$. Thus by linearity of expectation the expected number of cards given to Player $2$ is

$$ 4\sum_{k=1}^{13}\frac1{4k+1}-\frac1{4\cdot2+1}+\frac1{4\cdot2}=\frac{1431112832425163}{499934366731800}\approx2.8626\;, $$

where the two additional terms correct for the fact that the $3$ of hearts doesn't have to come before itself to be passed.

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