his problem set will refer to a standard deck of cards. Each card in such a deck has a rank and a suit. The 13 ranks, ordered from lowest to highest, are 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, and ace. The 4 suits are clubs (♣), diamonds (♢), hearts (♡), and spades (♠). Cards with clubs or spades are black, while cards with diamonds or hearts are red. The deck has exactly one card for each rank–suit pair: 4 of diamonds, queen of spades, etc., for a total of 13 · 4 = 52 cards. After binge watching her TV show, Sara still had some time to kill, so she pulled out a deck of cards and came up with a fun problem. Consider Sara’s standard deck of cards. You divide this deck in half by selecting uniformly at random 26 of the deck’s 52 cards and placing them in your left hand. You hold the remaining 26 cards in your right hand. Prove that the expected number of aces in your right hand is 2.
This problem set will refer to a standard deck of cards. Each card in such a deck has a rank and
a suit. The 13 ranks, ordered from lowest to highest, are 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen,
king, and ace. The 4 suits are clubs (♣), diamonds (♢), hearts (♡), and spades (♠). Cards with
clubs or spades are black, while cards with diamonds or hearts are red. The deck has exactly
one card for each rank–suit pair: 4 of diamonds, queen of spades, etc., for a total of 13 · 4 = 52
cards.
After binge watching her TV show, Sara still had some time to kill, so she pulled out a deck
of cards and came up with a fun problem. Consider Sara’s standard deck of cards. You divide
this deck in half by selecting uniformly at random 26 of the deck’s 52 cards and placing them
in your left hand. You hold the remaining 26 cards in your right hand. Prove that the expected
number of aces in your right hand is 2.
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