Question

A standard 52-card deck of French playing cards consists of four suits: hearts, spades, clubs, and diamonds. There are 13 cards of each suit; each suit has cards of rank 2 through 10, along with an ace, king, queen, and jack. Typically, hearts and diamonds are the red suits, while spades and clubs are the black suits.

Four cards are drawn from the deck, one at a time, without replacement.

a) The second card drawn is from a red suit. Based on this information, what is the probability it is a heart?

b) Calculate the probability of drawing exactly one heart (out of the four cards).

Answer 1

a) There are three cases:

i) Case1: If the first drawn card was a red heart. Then the probability the second card is a red heart given the card is red is 1/25

ii) Case 2: If the first drawn card was red but not heart. Then the probability the second card is a red heart given the card is red is 2/25

iii) Case3: If the first drawn card was not red. Then the probability that the second card was a red heart given the card is red is 2/26.

Thus the probability that the card is a heart given it is red=P(Red heart|case1)*P(Case1) + P(Red heart|case1)*P(Case1) +P(Red heart|case1)*P(Case1) = 1/25*2/52 + 2/25*24/52 + 2/26*1/2 = 99/1300 = 0.0761

b) There are four hearts in the deck. Thus number of ways drawing exactly 1 heart is .

While the total number of ways of selecting four cards are . Thus the probability of drawing exactly one heart is