lilmoore11p8

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 ?

Charlee Gentry

Expert

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:
$\mathbb{E}X=1+\mathbb{E}{X}_{1}+\mathbb{E}{X}_{2}+\mathbb{E}{X}_{3}+\mathbb{E}{X}_{4}=1+4\mathbb{E}{X}_{1}=1+4P\left({X}_{1}=1\right)=1+4\cdot \frac{3}{51}$
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: $P\left(A\right)=P\left(S\right)P\left(A\mid S\right)+P\left({S}^{\complement }\right)P\left(A\mid {S}^{\complement }\right)=\frac{1}{5}\cdot 1+\frac{4}{5}\frac{3}{51}$

Do you have a similar question?