Solved: In use the sample space given in Example. Write

Chapter 9, Problem 4E

(choose chapter or problem)

Problem 4E

In use the sample space given in Example. Write each event as a set, and compute its probability.

Example

Probabilities for a Deck of Cards

An ordinary deck of cards contains 52 cards divided into four suits. The red suits are diamonds (♦)and hearts (♥) and the black suits are clubs (♣)and spades (♠). Each suit contains 13 cards of the following denominations: 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king), and A (ace). The cards J, Q, and K are called face cards.

Mathematician Persi Diaconis, working with David Aldous in 1986 and Dave Bayer in 1992, showed that seven shuffles are needed to “thoroughly mix up” the cards in an ordinary deck. In 2000 mathematician Nick Trefethen, working with his father, Lloyd Trefethen, a mechanical engineer, used a somewhat different definition of “thoroughly mix up” to show that six shuffles will nearly always suffice. Imagine that the cards in a deck have become—by some method—so thoroughly mixed up that if you spread them out face down and pick one at random, you are as likely to get any one card as any other.

a. What is the sample space of outcomes?

b. What is the event that the chosen card is a black face card?

c. What is the probability that the chosen card is a black face card?

Solution

a. The outcomes in the sample space S are the 52 cards in the deck.

b. Let E be the event that a black face card is chosen. The outcomes in E are the jack, queen, and king of clubs and the jack, queen, and king of spades. Symbolically,

E = {J♣, Q♣, K♣, J♠, Q♠, K♠}.

c. By part (b), N(E)= 6, and according to the description of the situation, all 52 outcomes in the sample space are equally likely. Therefore, by the equally likely probability formula, the probability that the chosen card is a black face card is

Exercise

The event that the chosen card is black and has an even number on it.