CONDITIONAL PROBABILITY (PRACTICE PACKET)

Transcription

1 CONDITIONL PROILITY (PRCTICE PCKET) NME: PER; DTE: _ Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of these pairs of events are dependent? You flip a coin and get tails. You flip it a second time and get heads. You pull your friend's name out of a hat that holds 20 different names, replace the name, then draw out your friend's name again. C You spin a spinner divided into five equal parts and is numbered 1-5. You get a 3 on the first spin, and then spin again and get a 2 on the second spin. D You remove a black sock from a drawer without looking, then remove another black sock. 2. Which of these pairs of events are independent? You remove a blue glove from a drawer without looking, then remove another blue glove. You reach into a basket and draw a name for a prize, return the name into the basket, and then draw a second time for another prize. C Your CD player has a random mode that chooses songs randomly and plays each song once before repeating. While listening to the CD player in random mode, you hear track 5 first and then hear track 3 second. D You choose a member of the basketball team to be the center. You choose a different member to be a forward. 3. grab bag contains 7 football cards and 3 basketball cards. n experiment consists of taking one card out of the bag, then selecting another card. What is the probability of selecting a football card and then a basketball card if: (a) the first card is not replaced? (b) the first card is replaced? Express your answers in decimal form. If necessary, round answers to the nearest hundredth. (a) 0.23 C (a) 0.21 (b) 0.21 (a) 0.09 (b) 0.49 (b) 0.23 D (a) 0.49 (b) Find the probability of getting the results shown on the spinners. Express your answer as a fraction in simplest form. 1 C D Serita and Will are part of a group going to a basketball game. The group has 3 courtside seats, 5 upper level seats, and 2 seats in the mezzanine. Find the probability that both events and will occur if the group randomly chooses their seats. Express your answer as a percent. If necessary, round your answer to the nearest tenth. Event : Serita will sit courtside at the basketball game.

2 Event : Will will sit courtside at the basketball game. 30% C 9% 93.3% D 6.7% 6. If the spinner is spun twice, what is the probability that the arrow will stop on a vowel both times? C D 7. Which of the following scenarios does NOT describe independent events? Drawing two cards from a standard deck of cards that are both aces Rolling a fair number cube twice and getting 6 on both rolls C Flipping a fair coin twice and getting heads on both flips D Rolling a 3 on a fair number cube and flipping tails on a fair coin 8. Sales records at an automobile dealership show that they sold 40 red cars one month. Of those cars, 25 were new cars and 15 were used cars. James bought a red car from the dealership that month. What is the probability he bought a used car? Express your answer as a fraction in lowest terms. C D 9. In a bag of 20 candies, 12 are red and 15 have peanuts in them. If the events of picking a red candy and picking a candy with peanuts are independent, how many of the red candies have peanuts? 3 6 C 9 D Frieda made the following two-way table about the color and size of the marbles in a bag. lue Red Clear Total Small Large Total What is the probability that one of Frieda s marbles is red, given that it is large? C D 0.157

3 11.Tanisha made the following two-way table about the color and pattern of the socks she owns. White Yellow Red Total Patterned Solid Total What is the probability that one of Tanisha s socks is patterned, given that it is red? C D Three players are playing a game to decide who wins a prize. Which method of choosing a winner results in a fair game? Each player makes and throws a paper airplane. The player whose airplane travels the farthest is the winner. One player rolls a number cube. If the number that comes up is a 1 or a 2, the first player wins. If the number that comes up is a 3 or a 4, the second player wins. If the number that comes up is a 5 or a 6, the third player wins. C Each player rolls two number cubes. If the product of the numbers on the die is 1 to 12, the first player wins. If the product of the numbers is 13 to 24, the second player wins. If the product of the numbers is 25 to 36, the third player wins. D Three spinners, labeled,, and C, are numbered 1 through 8, 1 through 7, and 1 through 6, respectively. The sectors on each spinner have equal areas. The first player makes a 3-digit number C (with the number from spinner in the hundreds place, the number from in the tens place, and C in the ones place). The second player makes the 3-digit number C. The third player makes the 3-digit number C. The player whose number is greatest is the winner. 13. group of 12 friends playing a game wants to use a random number generator to decide who goes first. The generator will only create 2-digit numbers, from 00 to 99. Which method would produce a fair decision about who goes first? Let the numbers from 00 to 07 represent the first friend, 08 to 15 represent the second, and so on. Once all friends are assigned a group of numbers, leave the remaining numbers unassigned. Let 00 represent the first friend, 01 the second, and so on. Continue assigning numbers to friends until all 100 numbers are assigned. C Generate a number for the first friend. If it is between 00 and 11, that friend goes first. If not, repeat for the second friend, and so on. D Let the numbers from 00 to 11 represent the first friend, 12 to 23 represent the second, and so on until all 100 numbers are assigned. 14. bag of marbles contains 16 marbles, one of which is red. 4 friends are playing a game in which whoever gets the red marble wins. Which procedure does not make the game fair? Each friend takes 4 marbles from the bag and doesn t look at them. fter every player has 4 marbles, they all reveal their marbles simultaneously. Whoever has the red marble wins. The friends take turns picking single marbles from the bag. If a person gets the red marble, they win. If not, they put the marble they picked back in the bag and the game continues. C The friends take turns picking single marbles from the bag. If a person gets the red marble, they win. If not, they keep the marble they picked and the game continues. D The friends take turns picking from the bag, 4 marbles at a time. If a person gets the red marble, they win. If not, they keep the marbles they picked and the game continues.

4 15. Malcolm is playing a board game with 4 other players. He suspects that the game is not fair, because he played it 8 times without winning once. If the game is actually fair, what is the probability of this happening to Malcolm? If necessary, round your answer to the nearest hundredth of a percent % 0.39% C 62.5% D 0% 16. Which of these is not a fair game for two players? Toss a coin. If the coin lands heads, the first player wins. If it lands tails, the second player wins. Roll a number cube. If the number that comes up is prime, the first player wins. If the number that comes up is not prime, the second player wins. C Roll two number cubes, if the product of the two numbers is even, the first player wins. If the product is odd, the second player wins. D Toss two coins. If the coins come up with the same face, the first player wins. If the coins come up with different faces, the second player wins. 17. Suzi, Dmitri, and 11 of their classmates are playing a game. The game will have a single winner. If the game is fair, what is the probability that NEITHER Dmitri nor Suzi wins? C D 18. Melanie, ndrew, and 8 of their classmates are playing a game. The game will have a single winner. If the game is fair, what is the probability that ndrew wins? C D 19. In a game with players, what is the probability of NOT winning, assuming the game is fair?

5 C D 20. In a game, three players spin the spinners shown below. Player 1 wins if the sum of the two numbers is 2, 3, or 4, player 2 wins if the sum of the two numbers is 5, 6, or 7, and player 3 wins if the sum of the two numbers is 8, 9, or 10. Is the game a fair game? Why? Yes, because each player has an equal probability of winning No, because player 1 has a greater probability of winning C No, because player 2 has a greater probability of winning D No, because player 3 has a greater probability of winning 21. In a game show, a contestant has 3 questions to answer in 3 different categories. (She does not lose any money for wrong answers.) The Science question is worth $1000, and she has a 30% chance of answering incorrectly without help. The History question is worth $1500, and she has a 40% chance of answering incorrectly without help. The Math question is worth $2000, and she has a 20% chance of answering incorrectly without help. She can ask for help on two of the three questions. sk the Expert reduces her chance of answering any question incorrectly to 5%. Simplify It cuts her probability of answering any question incorrectly in half. To maximize her expected winnings, on which question should the contestant use sk the Expert? On which question should she use Simplify It? What are her expected winnings if she uses this strategy? Use sk the Expert on the Science question and Simplify It on the History question. Her expected winnings are $3750. Use Simplify It on the History question and sk the Expert on the Math question. Her expected winnings are $3800. C Use sk the Expert on the History question and Simplify It on the Math question. Her expected winnings are $3925. D Use Simplify It on the Science question and sk the Expert on the History question. Her expected winnings are $ One Computers is planning to send 2500 laptop computers to a huge computer store chain, Megatronix, which sells the -One computers to the public. If Megatronix customers return more than 12 of these computers because of defects, Megatronix will stop buying -One computers, which would be a financial disaster for -One.

6 The -One quality control group inspects 500 computers and finds that 498 of them have no defects. Predict the number of computers in the 2500 that are likely to have defects. ased on these results, should -One send the computers to Megatronix? 2; -One should send the computers, since there are likely to be fewer than 12 defective computers. 2490; -One should not send the computers, since there are likely to be more than 12 defective computers. C 10; -One should send the computers, since there are likely to be fewer than 12 defective computers. D 498; -One should not send the computers, since there are likely to be more than 12 defective computers. 23. Four coworkers are doing data-entry processing. lice processes 25% of the data, and 99% of her work is accurate. en processes 30% of the data, and 98% of his work is accurate. Cassie processes 31% of the data, and 97% of her work is accurate. Dennis processes the remaining data, and 96% of his work is accurate. customer informs the supervisor about a single inaccurate entry, and the supervisor decides Dennis is at fault because he has the highest error rate. Is this a good decision? No. lice is most likely at fault. No. en is most likely at fault. C No. Cassie is most likely at fault. D Yes. Dennis is most likely at fault. 24. The table below shows the results of an allergy test on a group of people. ased on these results, a doctor decides to prescribe allergy medication to every patient who tests positive. Is this a good decision? Why or why not? Has allergy Does not have allergy Total Tests positive Tests negative Total Yes, because 70% of the patients who have the allergy tested positive for the allergy Yes, because 11.5% of the patients tested positive for the allergy C No, because 70% of the patients who have the allergy tested positive for the allergy D No, because about 70% of the patients who tested positive don t have the allergy 25. Two factories make identical products. Of the items produced at factory, 10% have a particular defect, and of the items produced at factory, 30% have the same defect. ased on the table, which of the following statements is correct? Factory Factory Total Has defect No defect Total

7 If a randomly selected item has the defect, it likely came from factory. If a randomly selected item has the defect, it likely came from factory. C If a randomly selected item came from factory, it will likely have the defect. D If a randomly selected item does not have the defect, it likely came from factory. Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. spinner numbered 1 through 6 has sections that are red, yellow, and blue. Juan spins the spinner 50 times and records the results in the table below. ccording to the data, which events are independent? Even Odd Total Red Yellow lue Total Landing on a red section and landing on an even number Landing on a yellow section and landing on an even number C Landing on a blue section and landing on an even number D Landing on a red section and landing on an odd number E Landing on a yellow section and landing on an odd number F Landing on a blue section and landing on an odd number 2. For two events and, what must be true in order for and to be independent? C D E F 3. ngela has a pack of 40 cards: some red, some blue, some with letters, and some with numbers. ased on the data shown in the table, which statements are true? Red lue Total Number Letter Total

8 C D E F Picking a numbered card and picking a red card are independent events. Picking a lettered card and picking a blue card are independent events. 4. The 10 members of a club cannot agree on who should be president, and have decided to pick a member at random. Which of the following are fair techniques for picking a president? Each member s name is written on a slip of paper and one slip of paper is randomly picked out of a hat. The member whose name is picked is the president. The members take turns flipping a fair coin. The first member whose coin lands on heads is the president. C Each member is assigned a digit from 0 to 9 and a random number generator is used to pick one digit. The member whose number is picked is the president. D Each member picks a card from a standard deck of cards and one of the cards is randomly selected from among them. The member whose card is picked is the president. E Each member is assigned a number from 1 to 10 and a number cube is rolled twice. The member whose number is the sum of the digits is the president % of a group of 10,000 mice in a laboratory are known to have a particular genetic defect. The table below shows the results of a test for the defect. Which of the following are true? Defect No defect Total Test positive Test negative Total ,000 The test correctly identified 9.6% of the mice that have the defect. The test correctly identified 94% of the mice that do not have the defect. C The majority of mice that test positive actually have the defect. D The majority of mice that test negative do not have the defect. E bout 84.6% of the mice do not have the defect. Numeric Response 1. grab bag contains 3 football cards and 7 basketball cards. n experiment consists of taking one card out of the bag, replacing it, and then selecting another card. Find the probability of selecting a football card and then a basketball card. Express your answer as a decimal.

9 2. Jolene's mother has a job jar that contains 4 slips of paper, each with a different household chore written on it. One of the chores is washing the dishes. Daily chores are assigned by having each family member randomly select a slip of paper from the jar. fter the chores for the day are assigned, all of the slips of paper are placed back into the jar for the next day. If Jolene gets to draw first each day, what is the probability that she will be assigned to wash the dishes on Monday and Tuesday? Write your answer as a fraction. 3. Sheryl and Tess play basketball for the Tigers. The probability of Sheryl making a free throw is 0.25 and the probability of Tess making a free throw is Find the probability of Sheryl making a free throw and Tess not making a free throw. Write your answer as a decimal. 4. The table shows the books in na s bookcase. If she takes a paperback book from her shelf without looking, find the probability that it is a mystery. Express your answer as a decimal rounded to the nearest hundredth. Paperback Hardbound Nonfiction Mystery Romance Short nswer 1. You randomly draw letter tiles from a bag containing the letters from the word PENNSYLVNI. Find the probability. Then tell whether the events are independent or dependent. You draw an N from the bag, Then, without replacing the first N, you draw another N. 2. You flip a coin. Then you flip the coin again. re the two events independent or dependent? 3. ccording to a local meteorologist, there is a 60% chance of rain today. Historical data reveals that this meteorologist forecasts rain correctly 75% of the time. What is the probability of it raining today? 4. Monica has a bag with 30 marbles. 5 of the marbles are blue and striped, 10 of the marbles are striped but not blue, and 5 of the marbles are blue but not striped. Use probability to decide if picking a blue marble and picking a marble that has stripes are independent events. 5. The probability distribution table represents people who ran an obstacle course race, and whether they were members of a morning workout club. Finished Didn t Finish Totals Members a b c Non-members d f g Totals h j k Part : Write an expression to represent the probability of randomly choosing a participant who finished given that she was a non-member. Then write an expression to represent the probability of randomly choosing a participant who was a member, given that he finished.

10 Part : Suppose is not equal to. What does this mean about the independence or dependence of the event is a member of the club and the event finished the race? Explain. 6. The table represents the number of people from a neighborhood who voted for 1 of 3 candidates for state representative. Each voter was registered as Democrat, Republican, or Other. O'rien Myers Stevens Democrat Republican Other Part :. What is the probability of randomly choosing a voter who voted for Myers, given that the voter is a Republican? Round your answer to the nearest hundredth. Part : Do you think Myers is a Republican candidate? Explain why or why not. 7. Events and are independent if. Use this fact to show that if events and are independent, and. 8. drawer contains 10 red socks, 6 white socks, and 4 blue socks. Without looking, you draw out a sock, return it, and draw out a second sock. What is the probability that the first sock is red and the second sock is white? Round your answer to the nearest hundredth. 9. Mandy and 5 of her friends are playing a board game. One of the friends argues that the game is not fair because they have played 7 times and he has yet to win. If the game is actually fair, what is the probability of not winning 7 games in a row? If necessary, round your answer to the nearest tenth of a percent. 10. The spinner shown is divided into equal sections. In a game, 2 players each spin the spinner and record the sum of the results. Player wins if the sum is even, and player wins if the sum is odd. Show that this game is fair. Describe another way to assign the sums to the players that results in a fair game Problem 1. Elena chooses one of 100 tiles numbered from 1 to 100. Let event be the event that the number is even, let event be the event that the number is a multiple of 5, and let event be the event that the number is greater than 50. Find each probability. a. b.

11 c. d. e. re events and independent? re events and independent? Explain. 2. movie theater tracks the ticket and snack sales for one particular showing. 84 people who bought a ticket at a discounted rate bought a snack. 28 people who bought a ticket at a discounted rate did not buy a snack. 126 people who bought a regular price ticket bought a snack. 42 people who bought a regular price ticket did not buy a snack. Is buying a snack independent of buying a regular price ticket? Use conditional probability to justify your answer. 3. company manufactures office furniture. During a board meeting one day, the chair that the president of the company is using breaks. Since the company manufactured the chair, the president is furious and calls the quality manager into the meeting to find out which parts supplier is responsible for this humiliating defect. The company uses three suppliers for the part that broke. Supplier provides that part 20% of the time, supplier 30% of the time, and supplier C the remainder. Results from continuous random sampling show that on average, 3% of parts from supplier, 2.5% of parts from supplier, and 1% of parts from supplier C are defective. Use this information to determine the probability that a defective part is from each supplier. Which supplier is most likely responsible for the defective part? Show your work. 4. It is estimated that 0.5% of a population has a particular virus. There are two tests for this virus. Test correctly identifies someone who has the virus 97% of the time and correctly identifies someone who does not have the virus 97% of the time. Test correctly identifies someone who has the virus 99% of the time and correctly identifies someone who does not have the virus 95% of the time. a. For each test, what percent of the people who test positive actually do not have the virus? Show your work, and round to the nearest percent. b. For each test, what percent of the people who test negative actually do have the virus? Show your work, and round to the nearest percent. c. If the virus is very contagious, which test would be better to prevent infected people from spreading the virus? Use the results from parts a and b to explain. d. If the side effects of the medicine used to fight the virus are substantial, which test would be better to avoid treating people who don t need to be treated? Use the results from parts a and b to explain. 5. survey is given to 1000 members of a particular gym. 70% of the people go to the gym at least 3 times a week. 70% of people who go to the gym at least 3 times a week said that they go to the gym 3 times a week. 65% of people who go to the gym less than 3 times a week said that they go to the gym less than 3 times a week. a. Complete the two-way table to organize the data. Said go Go to gym at least 3 times Go to gym fewer than 3 times Total

12 at least 3 times Said go fewer than 3 times Total b. The owner of the gym wants to use the results of the survey to offer a special incentive package to everyone who says they go to the gym fewer than 3 times a week. Is this a good decision? Explain. c. The owner of the gym also wants to use the survey to offer everyone who says they go to the gym more than 3 times a week a new rewards program. Is this a good decision? Explain. 6. store houses 60% of its merchandise in store, and the remaining 40% in a warehouse for online orders. The store would like to centralize its returns department. If all returns are sent to the store, some items would have to be shipped to the warehouse. If all returns are sent to the warehouse, some items would have to shipped to the store. In the last month, 30% of the items purchased in store were returned, and 25% of the items purchased online were returned. Complete the two-way table with the frequencies associated with 100 items. Would it be more efficient to send all of the returned merchandise to the store or to the warehouse? Use conditional probability to justify your answer. Store Online Total Returned Not returned Total 100

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