Question

A deck of cards contain 52 cards consisting of 4 suits (Clubs, Diamonds, Hearts and Spades) and each suit has 13 cards. Cards are drawn one at a time at random with replacement.

(a) What is the expected time to get a card of Hearts?
(b) What is the expected time to get 4 cards with different suits?

Hint: The waiting times between getting i cards of different suits and i + 1 cards of different suits are independent, where i = 1, 2, 3.

Answer 1

(a)

Probability to get a card of Hearts = 13/52 = 1/4

Let X be the number of cards drawn to  get a card of Hearts. Then X ~ Geometric(p = 1/4)

By Geometric distribution, expected time to get a card of Hearts = 1/p

= 1/ (1/4)

= 4

(b)

On first draw, we get a card of different suit.

Probability to get a card of different suit after that = 39/52 = 3/4

Let X1 be the number of cards drawn to get a card of different suit. Then X1 ~ Geometric(p = 3/4)

By Geometric distribution, E(X1) = 1/p = 1/(3/4) = 4/3

Probability to get a card of different suit after getting 2 suits = 26/52 = 1/2

Let X2 be the number of cards drawn to get a card of different suit, after getting 2 suits. Then X1 ~ Geometric(p = 1/2)

By Geometric distribution, E(X2) = 1/p = 1/(1/2) = 2

Probability to get a card of different suit after getting 3 suits = 13/52 = 1/4

Let X3 be the number of cards drawn to get a card of different suit, after getting 3 suits. Then X3 ~ Geometric(p = 1/4)

By Geometric distribution, E(X4) = 1/p = 1/(1/4) = 4

The total time to get 4 cards with different suits = 1 + X1 + X2 + X3

Expected time to get 4 cards with different suits = E(1 + X1 + X2 + X3)

= 1 + E(X1) + E(X2) + E(X3)

= 1 + 4/3 + 2 + 4

= 25/3

= 8.33