Question

Suppose you have a well-shuffled faced-down pile of poker cards: the Ace of Hearts, 2 of Hearts, 3 of Hearts, and 4 of Hearts. You start picking the cards from the top until both the Ace and the 2 are picked up. What is the expected number of cards you have to pick? (Enter your final answer in decimal format and round to three decimal places.)

Answer 1

3.5 cards

Explanation:

The sample space for picking cards until the A of hearts and the two of hearts are picked up is:

{A,2} {2,A}

{A,1,2} {A,3,2} {A,4,2} {2,1,A} {2,3,A} {A,4,A}

{A,1,3,2} {A,1,4,2} {A,3,1,2} {A,3,4,2} {A,4,1,2} {A,4,3,2}

{2,1,3,A} {2,1,4,A} {2,3,1,A} {2,3,4,A} {2,4,1,A} {2,4,3,A}

There are 20 possible outcomes

The probabilities of needing 2, 3 or 4 cards to draw both the ace and the two of hears are:

`P(2)=(2)/(20)=(1)/(10)\\P(3)=(6)/(20) =(3)/(10)\\P(4)=(12)/(20) =(6)/(10)`

The expected number of cards needed is:

`EV=2*P(2)+3*(P3)+4*P(4)\\EV=0.1*2+0.3*3+0.6*4\\EV=3.5\ cards`