Replacement card after discarding the lowest

Question

Treat the ranks of a standard $52$ -card deck as $1$ (Ace) through $13$ (King). Four cards are dealt at random. The lowest-ranked of these four is thrown away, and a fresh card is drawn at random from the $48$ cards still in the deck. What is the expected rank of this fresh card?

Short answer

$7$ . The new card is a uniformly random card from the 48 undealt ones, and by symmetry each of the 52 cards is equally likely to end up as that card. Discarding the hand's minimum changes nothing about this, so the mean rank is that of a full deck, $\tfrac{1+13}{2}=7$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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