136 Probabilities of Mutually Exclusive Events


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1 Determine whether the events are mutually exclusive or not mutually exclusive. Explain your reasoning. 1. drawing a card from a standard deck and getting a jack or a club The jack of clubs is an outcome that both events have in common. not mutually exclusive 2. adopting a cat or a dog There are no common outcomes, so the two events are mutually exclusive. 3. JOBS Adelaide is the employee of the month at her job. Her reward is to select at random from 4 gift cards, 6 coffee mugs, 7 DVDs, 10 CDs, and 3 gift baskets. What is the probability that an employee receives a gift card, coffee mug, or CD? Let event G represent receiving a gift card. Let event C represent receiving a coffee mug. Let event D represent receiving a CD. There are a total of or 30 items. 4. CLUBS According to the table, what is the probability that a student in a club is a junior or on the debate team? Since some juniors are on the debate team, these events are not mutually exclusive. Use the rule for two events that are not mutually exclusive. The total number of students is 100. Determine the probability of each event. 5. If you have a 2 in 10 chance of bowling a spare, what is the probability of missing the spare? Let event A represent bowling a spare. Then find the probability of the complement of A. The probability of missing the spare is or 80%. esolutions Manual  Powered by Cognero Page 1
2 6. If the chance of living in a particular dorm is 75%, what is the probability of living in another dorm? Let event A represent the chance of living in a particular dorm. Then find the probability of the complement of A. 7. PROM In Armando s senior class of 100 students, 91 went to the senior prom. If two people are chosen at random from the entire class, what is the probability that at least one of them did not go to prom? You know that 91% of students went to the senior prom. The phrase at least one means one or more. So, you need to find the probability that either the first student chosen did not go to prom the second student chosen did not go to prom both students chosen did not go to prom. The probability of living in another dorm is 25%. or The complement of the event described above is the event that both students go to prom. Find the probability of this event, and then find the probability of its complement. Let event A represent choosing a student who goes to prom. Let event B represent choosing a student who geso to the prom after the first student has already been chosen. These are two dependent events, since the outcome of the first event affects the probability of the outcome of the second event. So, the probability that at least one of the students did not go to prom is about 17.3%. esolutions Manual  Powered by Cognero Page 2
3 Determine whether the events are mutually exclusive or not mutually exclusive. Then find the probability. Round to the nearest tenth of a percent, if necessary. 8. drawing a card from a standard deck and getting a jack or a six Since these two events cannot happen at the same time, these are mutually exclusive. Let event J represent getting a jack from a standard deck. Let event S represent getting a six from a standard deck. There are a total of 52 cards in the deck. 10. selecting a number at random from integers 1 to 20 and getting an even number or a number divisible by 3 18 is between 1 and 20 and is both even and divisible by 3. Since these two events can happen at the 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 rolling a pair of dice and getting doubles or a sum of 8 If you have the outcome (4, 4), it is both doubles and the sum is 8. Since these two events can happen at the same time, these are not mutually exclusive. Use the rule for two events that are not mutually exclusive. The total number of possible outcomes when rolling a pair of dice is tossing a coin and getting heads or tails Since these two events cannot happen at the same time, these are mutually exclusive. Let event T represent getting tails. Let event H represent getting heads. 12. drawing an ace or a heart from a standard deck of 52 cards Since these two events can happen at the same time, these are not mutually exclusive. Use the rule for two events that are not mutually exclusive. esolutions Manual  Powered by Cognero Page 3
4 13. rolling a pair of dice and getting a sum of either 6 or 10 Since these two events cannot happen at the same time, they are mutually exclusive. The total number of possible outcomes when rolling a pair of dice is CCSS MODELING An exchange student is moving back to Italy, and her homeroom class wants to get her a going away present. The teacher takes a survey of the class of 32 students and finds that 10 people chose a card, 12 chose a Tshirt, 6 chose a video, and 4 chose a bracelet. If the teacher randomly selects the present, what is the probability that the exchange student will get a card or a bracelet? Let event C represent getting a card. Let event B represent getting a bracelet. 14. SPORTS The table includes all of the programs offered at a sports complex and the number of participants aged What is the probability that a player is 14 or plays basketball? Determine the probability of each event. 16. rolling a pair of dice and not getting a 3 Since some 14 age 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. Let event C represent getting a 3 when rolling a pair of dice. Out of 36 outcomes, 11 of them have at least one 3 in it. The outcomes in which a 3 occurs are: (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (1, 3), (2, 3), (4, 3), (5, 3), and (6, 3). esolutions Manual  Powered by Cognero Page 4
5 17. drawing a card from a standard deck and not getting a diamond Let event D represent getting a diamond from a standard deck. 20. RAFFLE Namid bought 20 raffle tickets. If a total of 500 raffle tickets were sold, what is the probability that Namid will not win the raffle? 18. flipping a coin and not landing on heads Let event H represent landing on heads when flipping a coin. 19. spinning a spinner numbered 1 8 and not landing on JOBS Of young workers aged 18 to 25, 71% are paid by the hour. If two people are randomly chosen out of a group of 100 young workers, what is the probability that exactly one is paid by the hour? You know that 71% of the workers are paid by the hour. So, you need to find the probability that either the first worker chosen is paid by the hour or the second worker chosen is paid by the hour. Let event A represent choosing a worker who is paid by the hour and event B represent choosing another worker who is paid by the hour. These are two dependent events, since the outcome of the first event affects the probability of the outcome of the second event. Let event H represent landing on 5 when spinning a spinner numbered 1 8. So, the probability that exactly one is paid by the hour is about 42%. esolutions Manual  Powered by Cognero Page 5
6 22. RECYCLING Suppose 31% of Americans recycle. If two Americans are chosen randomly from a group of 50, what is the probability that at most one of them recycles? 26. The card is not a face card. King, Queen, and Jack are face cards. So, the number of face cards is 12. So, the probability that at most one of them recycles is about 90.4%. CARDS Suppose you pull a card from a standard 52 card deck. Find the probability of each event. 23. The card is a The card is red. 25. The card is a face card. King, Queen, and Jack are face cards. So, the number of face cards is 12. esolutions Manual  Powered by Cognero Page 6
7 27. MUSIC A school carried out a survey of 265 students to see which types of music students would want played at a school dance. The results are shown in the Venn Diagram. Find each probability. 28. CCSS CRITIQUE Tetsuya and Mason want to determine the probability that a red marble will be chosen out of a bag of 4 red, 7 blue, 5 green, and 2 purple marbles. Is either of them correct? Explain your reasoning. a. P(country or R&B) b. P(rock and country or R&B and rock) c. P(R&B but not rock) d. P(all three) a., and Therefore, neither student is correct. b. c. c. esolutions Manual  Powered by Cognero Page 7
8 29. CHALLENGE You roll 3 dice. What is the probability that the outcome of at least two of the dice will be less than or equal to 4? Explain your reasoning. 0.74; sample answer: Consider an outcome to be an event in which at least two of the dice show a 4 or less. There are three outcomes in which the values of two or more of the dice are less than or equal to 4 and one outcome where the values of all three of the dice are less than or equal to 4. You have to find the probability of each of the four scenarios and add them together. Let A be the first roll, B be the second roll, and C be the third roll.,, The sum of these probabilities is choosing a triangle that is equilateral and a triangle that is equiangular If the two events cannot happen at the same time, they are mutually exclusive. If a triangle is equilateral, it is also equiangular. The two can never be mutually exclusive. 32. choosing a complex number and choosing a natural number If the two events cannot happen at the same time, they are mutually exclusive. A natural number is also a complex number, so these events are not mutually exclusive. 33. OPEN ENDED Describe a pair of events that are mutually exclusive and a pair of events that are not mutually exclusive. If the two events cannot happen at the same time, they are mutually exclusive. If you pull a card from a deck, it can be either a 3 or a 5. The two events are mutually exclusive. If you pull a card from a deck, it can be a 3 and it can be red. The two events are not mutually exclusive. REASONING Determine whether the following are mutually exclusive. Explain. 30. choosing a quadrilateral that is a square and a quadrilateral that is a rectangle If the two events cannot happen at the same time, they are mutually exclusive. Since squares are rectangles, but rectangles are not necessarily squares, a quadrilateral can be both a square and a rectangle, and a quadrilateral can be a rectangle but not a square. They are not mutually exclusive. esolutions Manual  Powered by Cognero Page 8
9 34. WRITING IN MATH Explain why the sum of the probabilities of two mutually exclusive events is not always 1. Sample answer: When two events are mutually exclusive, it means that they can t both happen, but it does not mean that one or the other of the events must happen. The sum of all possible outcomes in a sample space must be 1. For example, if Event A and Event B are mutually exclusive, the sample space includes the probability of Event A, the probability of Event B, and the probability of neither Event A nor Event B, which must all sum to 1. The sum of the probabilities of Event A and Event B may be 1, but not necessarily. From a deck of cards, drawing a jack and drawing a queen are mutually exclusive events. However, a 3, 6, king, or many other cards can be drawn. 36. SHORT RESPONSE A cube numbered 1 through 6 is shown. If the cube is rolled once, what is the probability that a number less than 3 or an odd number shows on the top face of the cube? Possible outcomes: {1, 2, 3, 4, 5, 6} Number of possible outcomes: ALGEBRA What will happen to the slope of line p if it is shifted so that the yintercept stays the same and the xintercept approaches the origin? 35. PROBABILITY Customers at a new salon can win prizes during opening day. The table shows the type and number of prizes. What is the probability that the first customer wins a manicure or a massage? A B 0.35 C 0.5 D 0.65 F The slope will become negative. G The slope will become zero. H The slope will decrease. J The slope will increase. The slope of the line p is positive. If the yintercept stays the same and the xintercept increases, the slope of the line increases. So the correct choice is J. So, the correct choice is D. esolutions Manual  Powered by Cognero Page 9
10 38. SAT/ACT The probability of choosing a peppermint from a certain bag of candy is 0.25, and the probability of choosing a chocolate is 0.3. The bag contains 60 pieces of candy, and the only types of candy in the bag are peppermint, chocolate, and butterscotch. How many butterscotch candies are in the bag? A 25 B 27 C 30 D 33 E You roll a die and get a 2. You roll another die and get a 3. Since the probability of the fist event does not affect the probability of the second event, these are independent events. If two events A and B are independent, then P(A and B) = P(A)P(B). The correct option is B. Determine whether the events are independent or dependent. Then find the probability. 39. A king is drawn, without replacement, from a standard deck of 52 cards. Then, a second king is drawn. These events are dependent since the king is not replace before the second draw. If two events A and B are dependent, then P(A and B) = P(A)P(B A). esolutions Manual  Powered by Cognero Page 10
11 41. SPORTS A survey at a high school found that 15% of the athletes at the school play only volleyball, 20% play only soccer, 30% play only basketball, and 35% play only football. Design a simulation that can be used to estimate the probability that an athlete will play each of these sports. Some percentages, like 20%, 25%, 50%, 33.3%, are easy to represent with spinners or dice. When that is not the case, use a random number generator. Sample answer: Use a random number generator to generate integers 1 through 20 in which 1 7 represent football, 8 13 represent basketball, represent soccer, and represent volleyball. Do 20 trials, and record the results in a frequency table. Copy the figure and point P. Then use a ruler to draw the image of the figure under a dilation with center P and the scale factor r indicated. 42. r = Step 1: Draw rays from S though each vertex. Step 2: Locate A' on such that. Step 3: Locate B' on, C' on, and D' on in the same way. Then draw A'B'C'D'. The probability that an athlete plays only football is 0.35, only basketball is 0.30, only soccer is 0.25, and only volleyball is 0.1. esolutions Manual  Powered by Cognero Page 11
12 43. r = r = Step 1: Draw rays from P though each vertex. Step 1: Draw rays from P though each vertex. Step 2: Locate A' on such that. Step 2: Locate A' on such that. Step 3: Locate B' on and C' on. Then draw A'B'C'. Step 3: Locate B' on, C' on, D' on, E' on, and F' on. Then draw A'B'C'D'E'F'. esolutions Manual  Powered by Cognero Page 12
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