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 ?

Question

Charlee Gentry · Accepted Answer

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