Question

5. A hand of five cards is drawn without replacement from a standard deck.

(a) Compute the probability that the hand contains both the king of hearts and the king of spades.

(b) Let X = the number of kings in the hand. Compute the expected value E(X). Hint: consider certain random variables X1, . . . , X4.

(c) Let Y = the number of “face” cards in the hand. Given is that E(Y ) = 15/13. Find the variance V ar(Y ). Hint: consider certain random variables Y1, . . . , Y12 and use your result from part (a).

Answer 1

5.

Number of possible combinations with 5 cards

(a)

Number of possible combinations with the king of hearts and the king of spades

So, required probability = 19600 / 2598960 = 0.007541478

(b)

Number of possible combinations with no king

Number of possible combinations with one king

Number of possible combinations with two kings

Number of possible combinations with three kings

Number of possible combinations with four kings

Random variable X denotes number of kings in the hand.

So, the probability mass function is given by

(c)

King, queen and jack are face cards. So, there are 3 face caeds in each of 4 suits. So, we have 12 face cards in a deck.

Number of possible combinations with no face card

Number of possible combinations with one face card

Number of possible combinations with two face cards

Number of possible combinations with three face cards

Number of possible combinations with four face cards

Number of possible combinations with five face cards

Random variable Y denotes number of face cards in the hand.

So, the probability mass function is given by