From a deck of 52 cards, we select the ace of spades, plus 4 cards...

lilmoore11p8

lilmoore11p8

Answered

2022-07-06

From a deck of 52 cards, we select the ace of spades, plus 4 cards among the 51 remaining cards.
a) What is the expected number of aces among the five selected cards ?
b) We shuffle the five cards and pick one at random. What is the probability that it is an ace ?

Answer & Explanation

Charlee Gentry

Charlee Gentry

Expert

2022-07-07Added 19 answers

a) Number the 4 cards that are selected next to the ace of spades with 1,2,3,4.
Then let X i take value 1 if card i is an ace and take value 0 otherwise.
Then the number of selected aces is X := 1 + X 1 + X 2 + X 3 + X 4 and with linearity expectation and symmetry we find:
E X = 1 + E X 1 + E X 2 + E X 3 + E X 4 = 1 + 4 E X 1 = 1 + 4 P ( X 1 = 1 ) = 1 + 4 3 51
b) Let A denote the event that an ace is picked and let S denote the event that the ace of spades is picked.
Then probability that you pick an ace is: P ( A ) = P ( S ) P ( A S ) + P ( S ) P ( A S ) = 1 5 1 + 4 5 3 51

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