1. What is the probability of drawing each of the following cards from a standard playing deck?

Transcription

1 1. What is the probability of drawing each of the following cards from a standard playing deck? a. P(Jack) b. P(spade) c. P(Jack of spades) d. P(not spade) 2. When Ms. Shreve randomly selects a student in her class, she has a probability of of selecting a boy. a. If her class has 36 students, how many boys are in Ms. Shreve s class? b. If there are 11 boys in her class, how many girls are in her class? c. What is the probability that she will select a girl? 3. What is the probability of drawing each of the following cards from a standard playing deck? a. P(face card) b. P(card printed with an even number) c. P(red ace) d. P(purple card) 4. As Sandra drives, her music player randomly selects music from her playlist. Sandra s playlist contains: 3 traditional country songs 6 traditional rock songs 4 hip-hop rap songs 5 contemporary country songs 1 Latin rap song 3 traditional pop songs a. What is the probability that the player will select some rap music next? b. Find P(any traditional song) c. Find P(traditional pop song). d. Find P(not country song), the probability that the next song is not country music.

2 5. When you list all of the possible outcomes in a sample space by following an organized system (an orderly process), it is called a systematic list. There are different strategies that may help you make a systematic list, but what is most important is that you methodically follow your system until it is complete. To get home, Renae can take one of four buses: #41, #28, #55, or #81. Once she is on a bus, she will randomly select one of the following equally likely activities: listening to her MP3 player, writing a letter, or reading a book. (hint: sample space will have 12 outcomes) a. Create the sample space of all the possible ways Renae can get home and do one activity by making a systematic list. b. Use your sample space to find the following probabilities: i. P(Renae takes an odd-numbered bus) ii. P(Renae does not write a letter) iii. P(Renae catches the #28 bus and then reads a book) c. Does her activity depend on which bus she takes? Explain why or why not. 6. You are going to have the owner of the pet store randomly pick a fish for your aquarium at home. a. A tank at the pet store contains 9 spotted guppies, 14 red barbs, 10 red tetras, and 7 golden platys. What is the probability of getting a red colored fish from this tank? b. In a different tank that contains only golden platys, the probability of getting a female fish is 30%. If there are 18 female fish in the tank, how many total fish are in the tank? 7. Alexis, Bart, Chuck, and Darin all called in to a radio show to get free tickets to a concert. List all the possible orders in which their calls could have been received.

3 8. Can you bend your thumb backwards at the middle joint to make an angle, like the example below? Or does your thumb remain straight? The ability to bend your thumb back is thought to rely on a single gene. What about your tongue? Can roll your tongue into a U shape? Approximately of the U.S. population can bend their thumbs backwards and about can roll their tongues. One way to represent a sample space that has outcomes that are not equally likely is by using a probability area model. An area model uses a large square with an area of 1. The square is subdivided into smaller pieces to represent all possible outcomes in the sample space. The area of each outcome is the probability that the outcome will occur. For example, if of the U.S. population can bend their thumbs back, then the column representing this ability should take only one-fourth of the square s width, as shown below.

4 a. How should the diagram be altered to show that of the U.S. can roll their tongues? Copy this diagram on your paper and add two rows to represent this probability. b. The relative probabilities for different outcomes are represented by the areas of the regions. For example, the portion of the probability area model representing people with both special traits is a rectangle with a width of and a height of. What is the area of this rectangle? This area tells you the probability that a random person in the U.S. has both traits. c. What is the probability that a randomly selected person can roll his or her tongue but not bend his or her thumb back? Show how you got this probability. 9. Your teacher challenges you to a spinner game. You spin the two spinners with the probabilities listed below. The first letter comes from Spinner #1 and the second letter from Spinner #2. If the letters can form a two-letter English word, you win. Otherwise, your teacher wins. Make a probability area model of the sample space, and find the probability that you will win this game. Is this game fair? Why or why not?

5 10. Sinclair wonders how to model the spinner game in problem 9 using a tree diagram. He draws the tree diagram below. a. Sabrina says, That can t be right. This diagram makes it look like all the words are equally likely. Why is this tree diagram misleading? b. To make the tree diagram reflect the true probabilities in this game, Sabrina writes numbers on each branch showing the probability that the letter will occur. So she writes a on the branch for A, a on the branch for each T, etc. Following Sabrina s method, label the tree diagram with probabilities on each branch. c. According to the probability area model that you made in problem 9, what is the probability that you will spin the word AT? Now examine the bolded branch on the tree diagram shown above. How could the numbers you have written on the tree diagram be used to find the probability of spinning AT? d. Does this method work for the other combinations of letters? Similarly calculate the probabilities for each of the paths of the tree diagram. At the end of each branch, write its probability. (For example, write " " at the end of the "AT" branch.) Do your answers match those from problem 9? e. Find all the branches with letter combinations that make words. Use the numbers written at the end of each branch to compute the total probability that you will spin a word. Does this probability match the probability you found with your area model?

6 11. You and your best friend may not only look different, you may also have different types of blood! For instance, members of the American Navajo population can be classified into two groups: 73% percent (73 out of 100) of the Navajo population has type O blood, while 27% (27 out of 100) has type A blood. a. Suppose you select two Navajo individuals at random. What is the probability that both individuals have type A blood? This time, drawing an area model that is exactly to scale would be challenging. A probability area model (like the one above) is still useful because it will still allow you to calculate the individual areas, even without drawing it to scale. Copy and complete this generic probability area model. b. What is the probability that two Navajo individuals selected at random have the same blood type? 12. Zack and Nick (both from the U.S.) are shipwrecked on a desert island! Zack has been injured and is losing blood rapidly, and Nick is the only person around to give him a transfusion. Unlike the Navajo you learned of in problem 11, most populations are classified into four blood types: O, A, B, and AB. For example, in the U.S., 45% of people have type O blood, 40% have type A, 11% have type B, and 4% have type AB (according to the American Red Cross, 2004). While there are other ways in which people s blood can differ, this problem will only take into account these four blood types.

7 a. Make a probability area model representing the blood types in this problem. List Nick s possible blood types along the top of the model and Zack s possible blood types along the side. b. What is the probability that Zack and Nick have the same blood type? c. Luckily, two people do not have to have the same blood type for the receiver of blood to survive a transfusion. Other combinations will also work, as shown in the diagram below. Assuming that their blood is compatible in other ways, a donor with type O blood can donate to receivers with type O, A, B, or AB, while a donor with type A blood can donate to a receiver with A or AB. A donor with type B blood can donate to a receiver with B or AB, and a donor with type AB blood can donate only to AB receivers. Assuming that Nick s blood is compatible with Zack s in other ways, determine the probability that he has a type of blood that can save Zack s life! 13. Eddie is arguing with Tana about the probability of flipping three coins. They decided to flip a penny, nickel, and a dime. a. Make a sample space that shows all the possible outcomes. How many outcomes are there? b. Find the probability of each of the following events occurring. Be sure to show your thinking clearly: i. Three heads ii. One head and two tails iii. At least one tail iv. Exactly two tails

8 c. Which is more likely, flipping at least 2 heads or at least 2 tails? Explain. d. How would the probabilities change if Tana found out that Eddie was using weighted coins (coins that were not fair) so that the probability of getting heads for each coin was instead of? Would this change the sample space? Recalculate the probabilities in part (c) based on the new information. 14. There is a new game at the school fair called Pick a Tile, in which the player reaches into two bags and chooses one square tile and one circular tile. The bag with squares contains three yellow, one blue, and two red squares. The bag with circles has one yellow and two red circles. In order to win the game (and a large stuffed animal), a player must choose one blue square and one red circle. Gerri suggested making a systematic list of all the possible color combinations in the sample space, listing squares first then circles: RY BY YY RR BR YR So, says Gerri, the answer is. That doesn t seem quite right, says Marty. There are more yellow squares than blue ones. I don t think the chance of getting a yellow square and a red circle should be the same as getting a blue square and a red circle. a. Make a tree diagram for this situation. Remember to take into account the duplicate tiles in the bags. b. Find the probability of a player choosing the winning blue square-red circle combination. c. Should Gerri and Marty play this game? Would you? Why or why not?

9 d. Now draw a probability area model for the Pick a Tile game in problem 14. e. Use the probability area model to calculate the probability of each possible color combination of a square and a circular tile. 15. Rimshot McGee has a 70% free throw average. The opposing team is ahead by one point. Rimshot is at the foul line in a one-and-one situation with just seconds left in the game. (A one-and-one situation means that the player shoots a free throw. If they make the shot, they are allowed to shoot another. If they miss the first shot, they get no second shot. Each shot made is worth one point.) a. First, take a guess. What do you think is the most likely outcome for Rimshot: zero points, one point, or two points? b. Draw a tree diagram to represent this situation. c. Copy the probability area model below. Which part of the model represents Rimshot getting one point? How can you use the model to calculate the probability that Rimshot will get exactly one point? d. Use either your tree diagram or the area model to help you calculate the probabilities that Rimshot will get either 0 or 2 points. What is the most likely of the three outcomes?

10 16. When he was in first grade, Harvey played games with spinners. You were allowed to move if your color came up on both spinners. a. Harvey always chose purple because that was his favorite color. What was the probability that Harvey could move his marker? b. Is the event that Harvey wins a union or an intersection of events? c. Was purple the best color choice? Explain. d. If both spinners are spun, what is the probability that no one gets to move because the two colors are not the same? 17. Avery has been learning to play some new card games and is curious about the probabilities of being dealt different cards from a standard 52- card deck. Help him figure out the probabilities listed below. a. What are P(king), P(queen), and P(club)? b. What is P(king or club)? How does your answer relate to the probabilities you calculated in part (a)? c. What is P(king or queen)? Again, how does your answer relate to the probabilities you calculated in part (a)? d. What is the probability of not getting a face card? Jacks, queens, and kings are face cards. 18. Kiyomi has 4 pairs of pants (black, peach, gray, and cream), and she has 5 shirts (white, red, teal, black, and lavender). a. If any shirt can be worn with any pair of pants, represent the sample space of all possible outfits with both a probability area model and a tree diagram. How many outfits does she own?

11 b. The closet light is burned out, so Kiyomi must randomly select a pair of pants and a shirt. What is the probability that she will wear something black? 19. In a certain town, 45% of the population has dimples and 70% has a widow s peak (a condition where the hairline above the forehead makes a V shape). Assuming that these physical traits are independently distributed, what is the probability that a randomly selected person has both dimples and a widow s peak? What is the probability that he or she will have neither? Use a probability area model or a tree diagram to represent this situation. 20. Assume that two standard dice are being rolled. Let event A = {the sum is a multiple of 3} and event B = {the sum is a multiple of 4}. The P(A) = and the P(B) =. a. How many outcomes are in the intersection of events A and B? b. What is P(A or B)? 21. In a standard deck of 52 playing cards, 13 cards are clubs, and 3 of the clubs are face cards (K, Q, J). What is the probability of drawing one card that is: a. A club or a face card? Is this a union or an intersection? b. A club and a face card? Is this a union or an intersection? c. Not a club and not a face card? 22. Kayla brought snacks for her and her partner on the volleyball team. She packed flavored water (2 berry and 4 citrus), fruit (5 apricots, 2 apples, and 3 bunches of grapes), and small packages of crackers (2 regular and 2 whole wheat). Kayla will randomly choose one flavored water, one fruit, and one package of crackers. Show all the possible combinations of three snacks that Kayla could choose. What is the probability that Kayla will choose a high-fiber snack (any combination that includes both an apple and whole-wheat crackers).

12 23. Maribelle is playing the board game Eight The Hard Way with her friends. Each player rolls two dice on their turn, and moves according to the sum on the dice. However, if a player rolls two fours (called eight the hard way ), they instantly win the round of play and a new round is started. Shayna stepped into the kitchen to get snacks when she heard Maribelle shout Wow! I got an eight! Shayna knows Maribelle got an eight. Help Shayna investigate the probability that Maribelle rolled two fours and won the round of play. In other words calculate the conditional probability that Maribelle rolled two fours, given that you know she already rolled a sum of eight. a. Use an area model to represent all of the possible sums of numbers when rolling two dice. b. However, since you know that Maribelle rolled a sum of eight, the sample space is changed. Now the sample space is only all the ways a sum of eight can be rolled. On your area model, shade all of the ways a sum of eight can be rolled. How many different ways can a sum of eight be rolled, that is, how many outcomes are in the new sample space? c. You are interested in the event {eight the hard way}. How many different ways can two fours be rolled? d. What is the probability of the event {eight the hard way} given that you know Maribelle already rolled a sum of eight? e. Becca rolls high (meaning that she rolled a sum of nine or more). What is the conditional probability that she rolled an odd number, given that you know she rolled high?

13 24. At Einstein Technical University (ETU), data on engineering majors was collected: a. What is the probability of a student living on campus at ETU? b. Copy the table and shade the cells with engineering majors. What is the conditional probability of a student living on campus, given that you know a student is an engineering major? c. Two events, A and B, are independent if knowing that B occurred does not change the probability of event A occurring. That is, two events, A and B, are independent if P(A given B) = P(A). Are the events {live on campus} and {engineering} independent? d. Two events are mutually exclusive (or disjoint) if they cannot both occur at the same time. That is, two events are mutually exclusive if P(A and B) = 0. Are the events {on campus} and {engineering} mutually exclusive? 25. At Digital Technical Institute, the following data was collected: a. Are the events {live on campus} and {engineering majors} independent at this institute? b. Are the events {live on campus} and {engineering} mutually exclusive at this institute? What outcomes are in the intersection of {live on campus} and {engineering}?

14 26. In the children s game, Build-a-Farm, each player first spins a spinner. Half of the time the spinner comes up red and half of the time the spinner comes up blue. If the spinner is red, the player reaches into the red box. If the spinner is blue, the player reaches into the blue box. The red box has 10 chicken counters, 10 pig counters, and 10 cow counters, while the blue box has 5 chicken counters, 4 pig counters, and 1 cow counter. a. Draw a tree diagram, including probabilities, to represent the sample space for this game. b. What is the probability of getting a cow counter in one turn? Even though the events {spin} and {animal counter} are not independent, a modified area model, as shown below, is possible for this situation. c. Copy the area model and shade the parts of the diagram that correspond to getting a cow counter. Using the diagram, verify that P(cow) is the same as the one that you found in part (b). d. Let s investigate the conditional probability that a child s spin was red, given that you know the child got a cow counter. Since you know that the child got a cow counter, the new sample space is limited to only the outcomes that contain cow which is the area that you shaded in part (c). Considering only the outcomes that contain cow, what is the conditional probability that a child spun red, knowing that the child got a cow counter? e. Using the method in part (d), find the conditional probability that if you got a pig counter, your spin was blue.

15 27. On the midway at the county fair, there are many popular games to play. One of them is Flip to Spin or Roll. First, the player flips a coin. If a head comes up, the player gets to spin the big wheel, which has ten equal sections: three red, three blue, and four yellow. If the coin shows a tail, the player gets to roll a cube with three red sides, two yellow sides, and one blue side. If the wheel spin lands on blue, or if the blue side of the cube comes up, the player wins a stuffed animal. a. Draw a modified area model to represent the sample space for Flip to Spin or Roll. Note that the rectangles for heads will have different areas than the rectangles for tails. b. Suppose that you know that Tyler won a stuffed animal. Discuss this with your team and then shade the appropriate parts of the modified area model to help you figure out the probability that he started off by getting a head. 28. Raul is conducting a survey for the school news blog. He surveyed 200 senior-class students and found that 78 students had access to a car on weekends, 54 students had regular chores assigned at home, and 80 students neither had access to a car, nor had regular chores to do. Raul said he couldn t figure out how to put the data into a table like the one below. a. Copy and complete Raul s table to figure out the number of students in each cell. b. This type of table is called a two-way table and is often used to organize information and calculate probabilities. Two-way tables often include row and column totals also. If you have not already done so, add row and column totals to your two-way table. Is there an association between car privileges and having regular chores for this group? Explain your answer in the context of the problem.

16 29. There are 30 students in Mr. Cooper s class; 18 boys and 12 girls. Mr. Cooper chooses a student at random to take the attendance folder to the office. Four of the boys have previously taken the folder to the office, and 3 of the girls have previously taken it. a. Create a two-way table to display this data. b. If Mr. Cooper randomly selects a student, what is the probability he selects a boy who previously took the folder? Make a new two-way table, and fill in that probability. Then fill in the remaining cells with their respective probabilities. Include row and column totals. c. If a student is chosen at random, what is the probability that the student is a girl or is a student that has taken the folder previously? Use the probabilities from the table that you made in part (b). d. Shade the cells in your table from part (c) where a student has previously taken the folder. If a student previously took the folder, what is the probability that the student is a girl? 30. If Letitia studies for her math test tonight, she has an 80% chance of getting an A. If she does not study, she only has a 10% chance. Whether she can study or not depends on whether she has to work at her parents store. Earlier in the day, her father said there is a 50% chance that Letitia would be able to study. a. Draw a modified area model for the situation. b. Find the probability that Letitia gets an A on the math test. c. What are the chances that Letitia studied, given that she got an A? Show how you shaded the diagram. d. Create a two-way table that shows the probabilities for this situation. Include row and column totals. Verify using your table that if she studies, Letitia has an 80% chance of getting an A as described in the beginning of this problem.

17 31. At the University of the Great Plains the following data about engineering majors was collected: a. What is the conditional probability of living on campus, given that you know a student is an engineering major? b. Compare your answer to part (a) to the probability of living on campus. c. Are the two events, {living on campus} and {engineering major} independent? Use the probabilities to explain why or why not. 32. In a recent survey of college freshman, 35% of students checked the box next to Exercise regularly, 33% checked the box next to Eat five servings of fruits and vegetables a day, and 57% checked the box next to Neither. a. Create a two-way table to represent this situation. Include row and column totals. b. What is the probability that a freshman in this study exercises regularly and eats 5 servings of fruits and vegetables each day? c. What is the probability that a freshman in this study exercises regularly or eats 5 servings of fruits and vegetables each day? d. Do you think that freshmen who eat 5 servings of fruit and vegetables per day are more likely to exercise? In other words, are exercising and eating associated? e. Compare and contrast a two-way table with an area model.

18 33. Double Spin. The player gets to spin a spinner twice, but only wins if the same amount comes up both times. The $100 sector is of the circle. Nick is currently playing the game. a. Make an area model to show the sample space of every possible outcome for two spins. What is the probability that Nick wins? b. When Nick came home from the fair, he told Zack that he had won some money in the Double Spin game. Knowing that Nick won some money, what are the chances that he won $100? c. Make a two-way table that shows the probabilities for the Double Spin game. How does your table compare to the area model from part (a)? Explain. d. A mathematical way to express the conditional probability relationship is: e. This relationship is called the Multiplication Rule. Verify your answer to part (b) using the Multiplication Rule. Be sure to define events A and B. 34. To learn how the Multiplication Rule got its name, rewrite the Multiplication Rule starting with P(A and B) =. a. Write the relationship between event A and event B for when they are independent using symbols b. Substitute the independence relationship from part (b) into the Multiplication Rule that you wrote for part (a) to get another definition for independence.

19 35. The Laundry Shop sells washers and dryers. The owner of the store, Mr. McGee, thinks that a customer who purchases a washer is more likely to purchase a dryer than a customer that did not purchase a washer. He analyzes the sales from the last month and finds that a total of 240 customers made purchases. He counts 180 washers that were purchased and 96 dryers that were purchased. Mr. McGee then counts the number of sales that included both a washer and a dryer and finds 72 customers purchased both. Is there an association between the purchase of washers and dryers? Explain and show your reasoning using the relationships that you have learned in this lesson. 36. It is generally assumed that there is no relationship between height and IQ (a measure of intelligence). Thus, the heights for 175 randomly selected people are independent of their IQs. Using this assumption, complete the two-way table below. 37. A spinner has just two colors, red and blue. The probability the spinner will land on blue is x. a.what is the probability it will land on red? b.sketch an area model for spinning this spinner twice. c.when the spinner is spun twice, what is the probability that it will land on the same color both times?

20 38. A technology group wants to determine if bringing a laptop on a trip that involves flying is related to people being on business trips. Data for 1000 random passengers at an airport was collected and summarized in the table below. a. What is the probability of traveling with a laptop if someone is traveling for business? b. Does it appear that there is an association between bringing a laptop on a trip that involves flying and traveling for business? 39. In a certain small town, 65% of the households subscribe to the daily paper, 37% subscribe to the weekly local paper, and 25% subscribe to both papers. a.make a two-way table to represent this data. b.if a household is selected at random, what is the probability that it subscribes to at least one of the two papers? Shade these areas in your table. c.charlie s neighbor subscribes to a paper. What is the probability that he receives the daily paper? 40. Marcos is selecting classes for next year. He plans to take English, physics, government, pre-calculus, Spanish, and journalism. His school has a six-period day, so he will have one of these classes each period. a. How many different schedules are possible? b. How many schedules are possible with first-period pre-calculus? c. What is the probability that Marcos will get first-period precalculus?

21 41. Parents keep telling their teens to turn down the music or turn off the computer when studying. But teens insist that these distractions actually help them study better! In order to put this argument to rest, a psychologist studied whether subjects were able to memorize 20 index cards while listening to loud music or studying in silence. The sixty subjects had these results: a. What is the probability that a randomly chosen subject is able to memorize the index cards? b. What is the probability that a music listener memorizes the index cards? c. According to the data from this study, is the ability to memorize independent of listening to loud music? 42. Marty and Gerri played Pick a Tile, in which the player reaches into two bags. One bag contains square tiles and the other circular tiles. The bag with squares contains three yellow, one blue, and two red squares. The bag with circles has one yellow and two red circles. In order to win the game (and a large stuffed animal), a player must choose one blue square and one red circle. a. Complete the two-way table below.

22 b. What is the probability of a player choosing the wining blue-red combination? c. When Marty pulled her hand out of the bag, Gerri squealed with delight because she thought she saw something blue. If it was something blue, what is the probability that Marty won a stuffed animal? 43. A survey of local car dealers revealed that 64% of all cars sold last month had a Green Fang system, 28% had alarm systems, and 22% had both Green Fang and alarm systems. a. What is the probability one of these cars selected at random had neither Green Fang nor an alarm system? b. What is the probability that a car had Green Fang and was not protected by an alarm system? c. Are having Green Fang and an alarm system disjoint (mutually exclusive) events? d. Use the alternative definition of independence to determine if having Green Fang is associated with having an alarm.