Statistics: How many cards does the student expect to draw until the ace of spades appears?

Question

A student draws cards from a standard deck of playing cards until the ace of

spades appears for the first time. After every unsuccessful draw, the student replaces

the card and shuffles the deck thoroughly before selecting a new card. How many

cards does the student expect to draw until the ace of spades appears?

Answer 1

Because the drawn card is replaced each time, the probability of drawing the ace of spades remains 1/52 for each draw. Therefore X, the total number of draws needed to draw the ace of spades, is a geometric random variable with parameter p=1/52. The expected value of any geometric random variable is 1/p. So we expect 52 draws on average.

Answer 2

At most 52

the probability is 1/52

as I have already explained