740 CHAPTER 11 Probability Did You Know? The Dead Man’s Hand James Butler “Wild Bill” Hickok (1837–1876), known as “the fastest gun in the west,” was a famous scout and federal marshal. Hickok was shot in the back and mortally wounded by Jack McCall while playing poker in a saloon in Deadwood, Dakota Territory (now South Dakota). McCall was hanged for his deed. The cards Hickok was holding when shot are shown here. Since that time, the two pairs, aces and eights, have become known as “the dead man’s hand” (see Exercise 37). Hickok is buried next to Martha Canary, better known as “Calamity Jane,” in a cemetery overlooking the town of Deadwood. Solution The order in which the 4 members are selected is not important. Therefore, we will work this problem using combinations. P(selecting 4 sophomores) number of ways to select any 4 sophomores number of ways to select any 4 club members = Since there are 7 sophomores, the number of ways to select any 4 sophomores is C 35. 7 4 = Since there are a total of 12 club members, the number of ways to select any 4 club members is C 495. 12 4 = P C C (selecting 4 sophomores) 35 495 7 99 7 4 12 4 = = = The probability that Joy randomly selects 4 sophomores is 7 99 . 7 Now try Exercise 9 Example 2 A Heart Flush A flush in the game of poker is five cards of the same suit (5 hearts, 5 diamonds, 5 clubs, or 5 spades). If you are dealt a five-card hand, determine the probability that you will be dealt a heart flush. Solution The order in which the five hearts are dealt is not important. Therefore, we may work this problem using combinations. P(heart flush) number of possible 5-card heart flushes total number of possible 5-card hands = Since there are 13 hearts in a deck of cards, the number of possible five-card heart flush hands is C 1287. 13 5 = The total number of possible five-card hands in a deck of 52 cards is C 2,598,960. 52 5 = P C C (heart flush) 1287 2,598,960 33 66,640 13 5 52 5 = = = The probability of being dealt a heart flush is , 33 66,640 or 0.000495. ≈ 7 Now try Exercise 15 Example 3 Streaming Movie Selection For movie night, the Brown family has narrowed down their streaming choices to the following 10 movies: the 4 Toy Story movies, the 3 Cars movies, and the 3 Monsters , Inc. movies. If the Browns randomly select 3 movies to watch, determine the probability that a) no Toy Story movies are selected. b) at least one Toy Story movie is selected. c) one movie from each franchise is selected. Solution a) If no Toy Story movies are to be selected, then only Cars and Monsters, Inc. movies must be selected. Six movies are either Cars or Monsters, Inc. Thus, the number of ways that 3 movies can be selected from the Cars or Monsters, Inc. movies is C6 3 . The total number of movie selections is C 10 3 . Therefore, P Toy Story C C (no movies) 20 120 1 6 6 3 10 3 = = =
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