12-6 Probability and the Addition Rule Determine whether the events are mutually 3. JOBS Adelaide is the employee of the month at her exclusive. Explain your reasoning. job. Her reward is to select at random from 4 gift 1. drawing a card from a standard deck and getting a cards, 6 coffee mugs, 7 DVDs, 10 picture frames, jack or a club and 3 gift baskets. What is the probability that Adelaide receives a gift card, coffee mug, or picture SOLUTION: frame? A jack of clubs is both a jack and a club, so the events are not mutually exclusive. SOLUTION: Let event G represent receiving a gift card. Let event ANSWER: C represent receiving a coffee mug. Let event D not mutually exclusive; A jack of clubs is both a jack represent receiving a picture frame. and a club. There are a total of 4 + 6 + 7 + 10 + 3 or 30 items. 2. adopting a cat or a dog SOLUTION: A cat cannot be a dog, and a dog cannot be a cat, so the events are mutually exclusive. ANSWER: mutually exclusive; A cat cannot be a dog, and a dog cannot be a cat. ANSWER: or about 67% 4. SPORTS CARDS Dario owns 145 baseball cards, 102 football cards, and 48 basketball cards. He selects a card at random to give to his brother. What is the probability that he selects a baseball or a football card? SOLUTION: There are 145 + 102 + 48 = 195 total cards. Let B represent baseball cards and F represent football cards. So, the probability that Dario selects a baseball or a football card is or about 84%. ANSWER: or about 84% eSolutions Manual - Powered by Cognero Page 1 12-6 Probability and the Addition Rule 5. CLUBS According to the table, what is the 6. KITTENS Ruby’s cat had 8 kittens. The litter probability that a student in a club is a junior or on the included 2 gray females, 3 mixed-color females, 1 debate team? gray male, and 2 mixed-color males. Ruby wants to keep one kitten. What is the probability that she randomly chooses a kitten that is female or gray? SOLUTION: Because some of Ruby's kittens are both gray and female, these events are not mutually exclusive. Use the rule for two events that are not mutually exclusive. SOLUTION: Because some juniors are on the debate team, these The total number of kittens is given as 8. events are not mutually exclusive. Use the rule for P(gray or female) = P(gray) + P(female) – P(gray two events that are not mutually exclusive. The total and female) number of students is 100. So, the probability that Ruby randomly chooses a kitten that is female or gray is or 75%. ANSWER: or 75% ANSWER: or about 44% eSolutions Manual - Powered by Cognero Page 2 12-6 Probability and the Addition Rule Determine whether the events are mutually 8. drawing a card from a standard deck and getting a exclusive. Then find the probability. Round to jack or a six the nearest tenth of a percent, if necessary. SOLUTION: 7. rolling a pair of dice and getting doubles or a sum of 8 Because these two events cannot happen at the same SOLUTION: time, these are mutually exclusive. If you have the outcome (4, 4), it is both doubles and the sum is 8. Because these two events can happen Let event J represent getting a jack from a standard at the same time, these are not mutually exclusive. deck. Let event S represent getting a six from a Use the rule for two events that are not mutually standard deck. exclusive. There are a total of 52 cards in the deck. The total number of possible outcomes when rolling a pair of dice is 36. ANSWER: ANSWER: not mutually exclusive; or 27.8% mutually exclusive; or 15.4% eSolutions Manual - Powered by Cognero Page 3 12-6 Probability and the Addition Rule 9. selecting a number at random from integers 1 to 20 11. drawing an ace or a heart from a standard deck of 52 and getting an even number or a number divisible by cards 3 SOLUTION: SOLUTION: Because these two events can happen at the same 18 is between 1 and 20, and is both even and divisible time, these are not mutually exclusive. Use the rule by 3. Because these two events can happen at the for two events that are not mutually exclusive. same time, these are not mutually exclusive. Use the rule for two events that are not mutually exclusive. Let e represent an even number and d represent divisible by 3. ANSWER: not mutually exclusive; or 30.8% ANSWER: 12. rolling a pair of number cubes and getting a sum of not mutually exclusive; or 65% either 6 or 10 SOLUTION: 10. tossing a coin and getting heads or tails Because these two events cannot happen at the same SOLUTION: time, they are mutually exclusive. Because these two events cannot happen at the same The total number of possible outcomes when rolling a time, these are mutually exclusive. pair of number cubes is 36. Let event T represent getting tails. Let event H represent getting heads. ANSWER: ANSWER: mutually exclusive; or about 22.2% mutually exclusive; 100% eSolutions Manual - Powered by Cognero Page 4 12-6 Probability and the Addition Rule 13. SPORTS The table includes all of the programs 14. MODELING An exchange student is moving back offered at a sports complex and the number of to Italy, and her homeroom class wants to get her a participants aged 14–16. What is the probability that a going-away present. The teacher takes a survey of player is 14 or plays basketball? the class of 32 students and finds that 10 people choose a card, 12 choose a T-shirt, 6 choose a video, and 4 choose a bracelet. If the teacher randomly selects the present, what is the probability that the exchange student will get a card or a bracelet? SOLUTION: Let event C represent getting a card. Let event B represent getting a bracelet. SOLUTION: Because some 14-year-old participants play basketball, these events are not mutually exclusive. Use the rule for two events that are not mutually exclusive. The total number of players is 300. ANSWER: or about 43.8% ANSWER: 15. Talia is playing a board game where rolling two dice 56% determines the number of spaces she moves. In Talia’s current position, she needs to roll at least a sum of 9 to win. What is the probability that Talia will win on her next turn? SOLUTION: There are a total of 36 possible outcomes when two dice are rolled. ANSWER: eSolutions Manual - Powered by Cognero Page 5 12-6 Probability and the Addition Rule 16. A bag contains six red coins labeled 1 through 6 and 20. The card is a spade or an ace. six green coins labeled 5 through 10. What is the probability of picking a coin labeled with a 5? SOLUTION: SOLUTION: There are a total of 12 possible outcomes, with only 2 favorable outcomes. ANSWER: So, . or about 31% There is a 1 in 6 chance of picking a coin labeled with 21. The card is a 5 or a prime number. a 5. SOLUTION: ANSWER: CARDS Suppose you pull a card from a ANSWER: standard 52-card deck. Find the probability of or about 31% each event. 17. The card is a 2 or a queen. 22. The card is red or an ace. SOLUTION: SOLUTION: ANSWER: or about 15% ANSWER: 18. The card is a diamond or a heart. or about 54% SOLUTION: NACHO CHIPS A restaurant serves red, blue, and yellow tortilla chips. The bowl of chips Gabriel receives has 10 red chips, 8 blue chips, and 12 yellow chips. Gabriel chooses a chip at ANSWER: random. Find each probability. 23. P(red or blue) or 50% SOLUTION: 19. The card is a 7 or a club. There are 30 total tortilla chips in the bowl. SOLUTION: ANSWER: ANSWER: or about 31% eSolutions Manual - Powered by Cognero Page 6 12-6 Probability and the Addition Rule 24. P(blue or yellow) 27. EDUCATION Max surveyed 200 students at his school to determine how many nights per week they SOLUTION: do homework. His results are shown in the table. There are 30 total tortilla chips in the bowl. ANSWER: 25. P(yellow or not blue) SOLUTION: "Not blue" means it can be either red or yellow. a. What is the probability that a randomly chosen student does homework at least 3 nights per week? There are 30 total tortilla chips in the bowl. b. What is the probability that a randomly chosen student does homework no more than 3 nights per week? SOLUTION: ANSWER: a. 26. P(red or not yellow) b. SOLUTION: "Not yellow" means it can be red or blue. ANSWER: There are 30 total tortilla chips in the bowl. a. b. ANSWER: eSolutions Manual - Powered by Cognero Page 7 12-6 Probability and the Addition Rule 28. TILES Kirsten and José are playing a game. REASONING Determine whether the following are Kirsten places tiles numbered 1 to 50 in a bag. José mutually exclusive. Explain. selects a tile at random. If he selects a prime number 30. choosing a quadrilateral that is a square and a or a number greater than 40, then he wins the game. quadrilateral that is a rectangle What is the probability that José will win on his first turn? SOLUTION: SOLUTION: If the two events cannot happen at the same time, There are 50 numbered tiles in all, with 15 prime they are mutually exclusive.