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, 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 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%

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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%

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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%

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9. selecting a number at random from integers 1 to 20 11. drawing an 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%

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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:

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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 . 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%

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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:

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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. numbers, 10 numbers greater than 40, and 3 numbers that are both prime and greater than 40. Because squares are rectangles, but rectangles are not necessarily squares, a quadrilateral can be a square and a rectangle, and a quadrilateral can be a rectangle but not a square. They are not mutually ANSWER: exclusive.

ANSWER: 29. ERROR ANALYSIS George and Aliyah are Not mutually exclusive; sample answer: Because determining the probability of randomly choosing a squares are rectangles, but rectangles are not blue or red marble from a bag of 8 blue marbles, 6 necessarily squares, a quadrilateral can be a square red marbles, 8 yellow marbles, and 4 white marbles. and a rectangle, and a quadrilateral can be a Is either of them correct? Explain. rectangle but not a square.

31. choosing a triangle that is equilateral and a triangle that is equiangular

SOLUTION: SOLUTION: If the two events cannot happen at the same time, Aliyah is correct. To find the probability of blue or they are mutually exclusive. red, the individual probabilities should be added because the events are mutually exclusive. If a triangle is equilateral, it is also equiangular. The two can never be mutually exclusive. ANSWER: Aliyah; to find the probability of blue or red, the ANSWER: individual probabilities should be added because the Not mutually exclusive; sample answer: If a triangle events are mutually exclusive. is equilateral, it is also equiangular. The two can never be mutually exclusive.

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32. choosing a complex number and choosing a natural 34. WRITING IN MATH Explain why the sum of the number probabilities of two mutually exclusive events is not always 1. SOLUTION: If the two events cannot happen at the same time, SOLUTION: they are mutually exclusive. Sample answer: When two events are mutually exclusive, it means that they can’t both happen, but it A natural number is also a complex number, so these does not mean that one or the other of the events events are not mutually exclusive. must happen. The sum of all possible outcomes in a sample space must be 1. For example, if event A and ANSWER: event B are mutually exclusive, the sample space Not mutually exclusive; sample answer: A natural includes the probability of event A, the probability of number is also a complex number. event B, and the probability of neither event A nor event B, all of which must sum to 1. The sum of the 33. OPEN-ENDED Describe a pair of events that are probabilities of event A and event B may be 1, but not mutually exclusive and a pair of events that are not necessarily. mutually exclusive.

SOLUTION: ANSWER: If the two events cannot happen at the same time, Sample answer: When two events are mutually they are mutually exclusive. exclusive, it means that they can’t both happen, but it does not mean that one or the other of the events If you pull a card from a deck, it can be a 3 or a 5. must happen. The sum of all possible outcomes in a The two events are mutually exclusive. If you pull a sample space must be 1. For example, if event A and card from a deck, it can be a 3 and it can be red. The event B are mutually exclusive, the sample space two events are not mutually exclusive. includes the probability of event A, the probability of event B, and the probability of neither event A nor ANSWER: event B, all of which must sum to 1. The sum of the Sample answer: If you pull a card from a deck, it can probabilities of event A and event B may be 1, but not be a 3 or a 5. The two events are mutually exclusive. necessarily. If you pull a card from a deck, it can be a 3 and it can 35. Cindy’s bowling records indicate that for any frame, be red. The two events are not mutually exclusive. the probability that she will bowl a strike is 30%, a spare 45%, and neither 25%. What is the probability that she will bowl either a spare or a strike for any given frame?

SOLUTION:

ANSWER:

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36. Visitors to a school carnival throw a dart at a 37. The spinner shown here is divided into 8 equal rectangular target in order to win prizes. The prizes sectors. Elliott spins the spinner one time. What is the are determined by the row and column in which the probability that the pointer lands on an odd number or dart lands, as shown in the diagram. Tamiko throws a a blue sector? dart that lands at random on the target. Which is closest to the probability that she wins 3 tickets or a T-shirt?

A

A 22% B 40% B C 54% D 72% C

SOLUTION: D

E So, the correct answer is choice D. SOLUTION: ANSWER: Because it is possible to land on a blue sector that is D an odd number on the same spin, these events are not mutually exclusive. Use the rule for events that are not mutually exclusive and the fact that there are 8 sectors of the spinner.

P(odd number or blue sector) = P(odd number) + P(blue sector) – P(odd number and blue sector)

= =

The probability that the spinner lands on an odd number or a blue sector is . So, the correct answer is choice D.

ANSWER: D

38. Chelsea has a piece of fabric with the dimensions shown below. She spreads out the fabric on a table and then accidentally lets a drop of ink fall onto the fabric. eSolutions Manual - Powered by Cognero Page 10 12-6 Probability and the Addition Rule

39. A single number cube is rolled. Find each probability.

a. P(3 or 5) b. P(at least 4)

SOLUTION: a.

b. There are 6 total possible outcomes, with 3 of them favorable: 4, 5, and 6. So, the probability of Assuming the ink lands at a random point on the rolling at least a 4 is . fabric, which is closest to the probability that it lands in the white row or checkerboard column? ANSWER: a. A 42% B 38% b. C 25% D 4%

SOLUTION: Because it is possible for the ink to drop on a piece of the fabric that is both white and checkerboard, these events are not mutually exclusive. The white section is represented by the area of a 36-by-5-inch rectangle. The checkerboard column is represented by the area of a 6-by-20 inch rectangle. Use the rule for events that are not mutually exclusive and the fact that the total area of the fabric is 36 · 20 or 720 square inches.

P(white or checkerboard) = P(white) + P(checkerboard) – P(white and checkerboard)

= = = =

The probability that Chelsea drops a spot of ink on a white row or the checkered column is or about 38%. So, the correct answer is choice B.

ANSWER: B

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40. MULTI-STEP The extracurricular activities in which members of the senior class at Valley View High School participate are shown in the Venn diagram.

a. How many students are in the senior class? b. How many seniors participate in athletics? c. If a senior is randomly chosen, what is the probability that he or she participates in athletics or drama? d. If a senior is randomly chosen, what is the probability that he or she participates in only drama and band?

SOLUTION: a. To find the total number of students in the senior class, add all values: 38 + 30 + 51 + 4 + 10 + 8 + 137 + 67 = 345.

b. To find the number of seniors who participate in athletics, add the values within the Athletics circle: 4 + 10 + 8 + 137 = 159.

c.

d.

ANSWER: a. 345 b. 159 c. d.

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