11.9 Solving Probability Problems by Using Combinations 743 29. Baseball Roster The Auburn University baseball roster consists of 12 pitchers, 2 catchers, 6 infielders, and 5 outfielders. If the team manager randomly selects 9 players, what is the probability that he selects 1 pitcher, 1 catcher, 4 infielders, and 3 outfielders? 144 81,719 30. Restaurant Staff The staff at The Captain’s Lair consists of 5 cooks, 11 servers, and 4 bartenders. If the restaurant manager randomly selects 10 staff members, what is the probability that she selects 2 cooks, 7 servers, and 1 bartender? 300 4199 31. A Royal Flush A royal flush consists of an ace, king, queen, jack, and 10 all in the same suit. If 7 cards are dealt from a standard 52-card deck, determine the probability of getting a a) royal flush in spades. 1 123,760 b) royal flush in any suit. 1 30,940 32. Poker Probability A full house in poker consists of three of one kind and two of another kind in a five-card hand. For example, if a hand contains three kings and two 5’s, it is a full house. If 5 cards are dealt from a standard 52-card deck, determine the probability of getting three kings and two 5’s. 1 108,290 33. Poker Hand Ronnie is playing poker and is dealt his hand of 5 cards from a standard 52-card deck. What is the probability that Ronnie is dealt 1 club, 1 diamond, 1 heart, and 2 spades? 2197 33,320 34. Poker Hand Bianca is playing poker and is dealt her hand of 5 cards from a standard 52-card deck. What is the probability that Bianca is dealt 2 face cards, 2 aces, and 1 card that is neither a face card nor an ace? 297 54,145 35. Jokers Included When a standard deck of cards is purchased, the deck usually includes 2 jokers. This means if the jokers are included, the deck contains 54 cards. Teresa is playing a game in which she is dealt 8 cards from a deck that includes the jokers. What is the probability that Teresa’s hand includes at least one joker? 44 159 B Christopher/Alamy Stock Photo 36. Euchre Hand A deck of cards for playing euchre contains 24 cards: 9, 10, jack, queen, king, and ace of each suit. Sergio is playing euchre and is dealt a 5-card hand. What is the probability that Sergio’s hand includes at least one ace? 1125 1771 37. “Dead Man’s Hand” A pair of aces and a pair of eights is often known as a “dead man’s hand.” (See the Did You Know? on page 740.) a) Determine the probability of being dealt a dead man’s hand (any two aces, any two eights, and one other card that is not an ace or an eight) when 5 cards are dealt, without replacement, from a standard 52-card deck. 33 54,145 b) The actual cards “Wild Bill” Hickok was holding when he was shot were the aces of spades and clubs, the eights of spades and clubs, and the 9 of diamonds. If you are dealt five cards, determine the probability of being dealt this exact hand. 1 2,598,960 38. Poker Straights A straight in poker is 5 cards of any suit in sequence. A jack is considered an 11, a queen is considered a 12, a king is considered a 13, and an ace is considered as either a 1 or a 14. The following are examples of straights: ace, 2, 3, 4, 5; 6, 7, 8, 9, 10; 8, 9, 10, jack, queen; 10, jack, queen, king, ace. Remember the suit does not matter when forming a straight. Molly is playing poker and is dealt 5 cards from a standard 52-card deck. a) What is the probability that Molly is dealt a straight consisting of an ace, a 2, a 3, a 4, and a 5? 64 162,435 b) What is the probability that Molly is dealt any straight? Hint: A straight may start with an ace, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Therefore, multiply the answer from part (a) by 10. 128 32,487 Challenge Problems/Group Activities 39. Selecting Officers A club consists of 15 people including Ali, Kendra, Ted, Alice, Marie, Dan, Linda, and Frank. From the 15 members, a president, vice president, and treasurer will be randomly selected. An advisory committee of 5 other individuals will also be randomly selected. a) Determine the probability that Ali is selected president, Kendra is selected vice president, Ted is selected treasurer, and the other 5 individuals named above form the advisory committee. 1 2,162,160 b) Determine the probability that 3 of the 8 individuals named are selected for the three officers’ positions and the other 5 individuals named are selected for the advisory board. 1 6435 40. Alternate Seating If three men and three women are to be randomly assigned to six seats in a row at a theater, determine the probability that they will alternate by gender. 1 10
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