Probability Rules

According to sales records at a local coffee shop, 75% of all customers like hot coffee, 30% like iced coffee, and 22% like both hot and iced coffee. The Venn diagram displays the coffee preferences of the customers. A randomly selected customer is asked if they like hot or iced coffee. Let H be the event that the customer likes hot coffee and let I be the event that the customer likes iced coffee. What is the probability that the customer only likes iced coffee? 0.08 0.17 0.22 0.53

a

Executives for a company that prints logos on products are expanding the company's services to include souvenirs such as hats, shirts, and foam fingers for sports teams. The data they collected from a sample of 300 adults about their favorite sport to watch and their favorite souvenir to buy are shown in the table. A survey participant is randomly selected. Let F be the event that the participant prefers football and let N be the event that the participant prefers the foam finger. What is the value of P(F and N)? 0.13 0.32 0.35 0.39

a

A deli owner made a probability distribution chart for the meat choices of their customers' sandwiches when the sandwiches contain only one meat. Ham Turkey Roast Beef Tuna Salami 0.28 0.31 0.19 ? 0.13 What is the missing probability in the table? 0 0.09 0.20 0.50

b

A group of ticket takers at a box office for a new theater noticed that in the first year of the theater's operation, the genre breakdown of the movies was 10% horror, 39% comedy, 28% drama, and 23% action. If a movie from the theater's first year of operation was selected at random, which of the following identifies the probability distribution for the movie's genre? ActionComedyDramaHorror0.100.390.280.23 ActionComedyDramaHorror0.230.390.280.10 ActionComedyDramaHorror0.390.230.100.28 ActionComedyDramaHorror0.390.100.230.28

b

Carlos thinks the traffic light to get out of his neighborhood is red more often than green. He decides to collect data to determine the probability of the light being red upon his approach. The graph of his long-run relative frequencies is shown. Which conclusion can be drawn from this graph? About half of the time, the traffic light is red when Carlos leaves his neighborhood. About 63% of the time, the traffic light is red when Carlos leaves his neighborhood. If the true probability that the traffic light is red when Carlos leaves his neighborhood is 0.63, there would be no variation in the graph. The probability that the traffic light is red when Carlos leaves his neighborhood cannot be determined from this graph because there is no pattern in a long series of traffic lights.

b

Executives for a car dealership are interested in the sales for the type of vehicle, SUV or truck, and the type of power train, two-wheel drive (2WD), four-wheel drive (4WD), or all-wheel drive (AWD). The data from the sales of 165 vehicles are displayed in the two-way table. A vehicle is randomly selected. Let T be the event that the vehicle is a truck and A be the event that the vehicle has all-wheel drive. What is the value of P(TC and AC)? 0.06 0.20 0.32 0.80

b

Executives for a company that prints logos on products are expanding the company's services to include souvenirs such as hats, shirts, and foam fingers for sports teams. The data they collected from a sample of 300 adults about their favorite sport to watch and their favorite souvenir to buy are shown in the table. A survey participant is randomly selected. Let S be the event that the participant prefers soccer and let T be the event that the participant prefers a T-shirt. What is the value of P(S and T)? 0.01 0.03 0.26 0.50

b

Shamir loves watching professional basketball. His favorite player, Freddy Rocket, successfully completes 85% of his free throws. During one game, Rocket misses his first four free throws. Shamir says that the next free throw has to be a success since Rocket rarely misses so many in a row. Is Shamir's reasoning correct? Yes, Rocket missed the first four free throws, so the fifth one is due to be a success. No, the probability of Rocket making a free throw is 0.85 over the long run. Yes, Rocket has a high probability of making free throws, so the next one must go in the basket. No, it is very unlikely for Rocket to miss so many free throws in a row, so Shamir should expect the player to continue missing baskets.

b

Ten people (labeled 1-10) have purchased raffle tickets for a fundraiser. However, they did not all purchase the same number of tickets. One ticket is to be selected at random. Which of the following could be the probability distribution for the winning ticket? 123456789100.100.100.100.100.100.100.100.100.100.10 123456789100.010.010.050.070.680.010.050.030.010.08 123456789100.010.110.020.120.030.130.040.140.050.15 123456789100.1250.1250.125-0.1250.1250.1250.1250.1250.1250.125

b

Three siblings, Peyton, Cameron, and Dakota, all ask their parents to borrow the family car for different events around town. Since they cannot all borrow the car at the same time, the parents decide to use randomness to decide who gets the car. They will roll a single, fair, six-sided number cube. Peyton gets the car if a 1 or 2 is rolled. Cameron gets the car if a 3 or 4 is rolled, and Dakota gets the car if a 5 or 6 is rolled. Which of the following is the probability distribution for who gets the car? PeytonCameronDakota 1 1 1 PeytonCameronDakota 1/3, 1/3, 1/3 PeytonCameronDakota 0.3, 0.3, 0.3 PeytonCameronDakota 1/6, 1/6, 1/6

b

Travel agents collected data from recent travelers about their modes of transportation for their vacations. They found that 37% traveled by airplane, 8% traveled by train, and 7% traveled by airplane and train. Let A be the event that the mode of travel was airplane and let T be the event that the mode of travel was train. What is the value of P(A and Tc), which is represented by 1 in the Venn diagram? 0.08 0.30 0.37 0.62

b

A researcher randomly surveyed 122 college professors to determine what types of courses they teach and their sleeping habits. The two-way table displays the data. Suppose a survey respondent is randomly selected. Let M = professor teaches math and B = professor is an early bird. What is the value of P(B|M)?

c

A survey of 500 college students moving into their dorm revealed that 425 brought a microwave, 380 brought a video game console, and 50 brought neither a microwave nor a game console. A survey participant is randomly selected. Let M be the event that the participant brought a microwave and let C be the event that the participant brought a video game console. Organize these events in a two-way table. What is the probability that the participant brought a microwave or a console, P(M or C)? 0.71 0.86 0.90 0.95

c

According to a recent survey of first-year high school students, 28% chew gum daily. The students were also asked if they had recently gotten a cavity filled at the dentist. Of the 47% of first-year students who responded that they had a cavity recently filled, only 39% chewed gum daily. Is chewing gum independent of having a cavity filled recently? Yes, P(Gum) = P(Gum|Cavity). Yes, P(Gum) = P(Cavity|Gum). No, P(Gum) ≠ P(Gum|Cavity). No, P(Gum) ≠ P(Cavity|Gum).

c

Reese, Greg, and Brad meet once a week for coffee. They each have their favorite café and, to be fair, they use randomization to choose where they will meet. Each person has a colored marble: red (R) for Reese, green (G) for Greg, and blue (B) for Brad. Each week, all three marbles are mixed well in a bag and a marble is selected. The favorite café of the person associated with the selected marble is chosen for that week's meeting. What is the probability that Reese will get to pick the café for at least one of the first two weeks?

c

Reese, Greg, and Brad meet once a week for coffee. They each have their favorite café and, to be fair, they use randomization to choose where they will meet. Each person has a colored marble: red (R) for Reese, green (G) for Greg, and blue (B) for Brad. Each week, all three marbles are mixed well in a bag and a marble is selected. The favorite café of the person associated with the selected marble is chosen for that week's meeting. Which of the following represents the sample space for choosing a café for the first two weeks? R & G, R & B, G & R, G & B, B & R, B & G R & R, R & G, R & B, G & G, G & B, B & B R & R, R & G, R & B, G & R, G & G, G & B, B & R, B & G, B & B R & R, R & G, R & B, R & G, G & R, G & G, G & B, B & R, B & G, B & B

c

Sports science researchers determined that, for those people who skateboard, 22% have never had an injury, 45% have had one injury, 18% have had two injuries, and 15% have had three or more injuries. What is the probability that a randomly chosen skateboarder has had one or two injuries? 0.18 0.45 0.63 0.78

c

Students majoring in psychology surveyed 200 of their fellow students about their dreams. The results of the survey are shown in the Venn diagram. Let B be the event that the participant dreams in black and white and let C be the event that the participant dreams in color. What is the probability that a randomly selected participant dreams in black and white or color? 0.06 0.07 0.13 0.26

c

A contractor claims that she finishes a job on time 90% of the time. Last month, she only completed 7 out of her 10 jobs on time. To see if this is surprisingly low, a simulation was conducted 100 times under the assumption that she really does complete 90% of her jobs on time. The dotplot contains 100 trials of this simulation. Based on this dotplot and the sample of last month's on-time completions, which conclusion can be drawn? The contractor's true, on-time completion rate is only 50%. It is most likely that the contractor will complete about 9 out of 10 jobs. If we used a larger sample size of 40 jobs, the simulated dotplot would be different; therefore, we cannot draw a conclusion. The dotplot does not provide convincing evidence that her true, on-time completion rate is less than 90% because 7 or fewer on-time completions happened 19% of the time in the simulation.

d

A large city's transit department claims that only 10% of city buses run off schedule. To test this claim, a random sample of 10 buses is chosen at random. Five of the buses are running off schedule. To see how unusual this sample of buses is, a simulation of 100 trials was conducted under the assumption that 10% of the buses run off schedule. Based on the dotplot of the simulation results and the sample of 10 buses, which conclusion can be drawn? The true probability that a city bus is running off schedule is 3%. If we continued to take more samples of 10 buses, the center of the distribution would shift to 1. It is most likely that exactly one out of 10 buses is running off schedule. There is about a 3% chance that 5 or more buses are running off schedule. This is unusual and is convincing evidence that the true probability that a bus is off schedule is more than 10%.

d

A popular board game has players twist their bodies around so that their hands and feet touch small colored dots. A spinner with equal areas for each body part (left hand, right hand, left foot, or right foot) is used. Which of the following is the correct probability distribution for the body part chosen to be placed? LHRHLFRF0.500.250.50-0.25 LHRHLFRF1111 LHRHLFRF4444 LHRHLFRF0.250.250.250.25

d

A researcher randomly asked 284 people how they prefer their eggs cooked and if they prefer orange juice or coffee with their breakfast. The two-way table displays the data. Suppose a survey respondent is randomly selected. Let event F = fried eggs and let event O = orange juice. What is the value of P(O|F)?

d

A teacher claims that there is a 50% chance that she will collect homework for a grade on any given day. One week, she collected all five daily homework assignments. A student in this class is upset and explains that the teacher should not collect any homework assignments the following week in order to honor her 50% probability claim. Is the student's reasoning correct? Yes, the teacher should not collect homework assignments next week to bring the probability of homework being collected back to 0.5. No, if the teacher collects homework five days in a row, it is not possible for the probability of homework being collected to be 0.5. Yes, it is unlikely that the teacher would randomly collect homework assignments five days in a row, so not having a homework collection next week is due to happen. No, collecting homework and not collecting homework are equally likely in the long run, so whether or not the teacher collects homework on any single day cannot be determined.

d

According to a recent survey of adults, 38% say the almond is their favorite nut. The adults were also asked where they lived. Of the 19% of those who responded that they live in California, 40% chose the almond as their favorite nut. Is liking almonds independent of residency? Yes, P(almonds) = P(California|almonds). Yes, P(almonds) = P(almonds|California). No, P(almonds) ≠ P(California|almonds). No, P(almond) ≠ P(almonds|California).

d

According to sales records at a local coffee shop, 75% of all customers like hot coffee, 30% like iced coffee, and 22% like both hot and iced coffee. The Venn diagram displays the coffee preferences of the customers. A randomly selected customer is asked if they like hot or iced coffee. Let H be the event that the customer likes hot coffee and let I be the event that the customer likes iced coffee. What is the probability that a randomly selected customer likes hot or iced coffee? 0.22 0.30 0.61 0.83

d

An animal researcher randomly selected 98 dogs and cats and recorded if they napped between 2:00 p.m. and 2:30 p.m. The two-way table displays the data. Suppose an animal is randomly selected. Let event C = cat and let event N = nap. What is the value of P(C|N)?

d

An online news report claims that 50% of online news readers work in the business industry. To test this claim, a researcher takes an SRS of 25 online news readers. Nine of them work in the business industry. A simulation of 65 trials was conducted under the assumption that 50% of online news readers really do work in the business industry. Based on this dotplot and the sample of 25 online news readers, which conclusion can be drawn? Since 0.36 of the sample works in the business industry, 0.36 is the true probability that an online news reader works in the business industry. It is most likely that, out of 25 online readers, between 12 and 13 work in the business industry. Because there appear to be outliers present that are greater than 16, we can conclude that more than 50% of online readers work in the business industry. There is about a 0.046 chance that 9 or fewer online readers work in the business industry. This is unusual and is convincing evidence that less than 50% of online readers work in the business industry.

d

For students majoring in Hospitality Management, it was determined that 5% have visited 1-10 states, 16% have visited 11-20 states, 45% have visited 21-30 states, 19% have visited 31-40 states, and 15% have visited 41-50 states. Suppose a Hospitality Management student is picked at random. What is the probability that the student has not visited between 21 and 30 states? 0.21 0.34 0.45 0.55

d

Forty percent of the beads in a bag of more than 10,000 beads are yellow. Suzy pulls out 10 beads, one at a time with replacement, and notes that eight of these beads are yellow. She says the next bead pulled out will not be yellow because a yellow bead has been pulled out too many times in a row. Is Suzy's reasoning correct? Yes, it is unlikely that she would pull out so many yellow beads, so the next bead cannot be yellow. Yes, Suzy has correctly decided that more than 40% of the beads in this large bag are yellow. No, if a yellow bead is pulled 8 out of 10 times, the bag of beads is unfairly weighted toward yellow. No, it is true that the probability of pulling a yellow bead is 0.40, but Suzy should not expect that exactly 40% of such a small number of beads pulled will be yellow.

d

Sports science researchers determined that, of those people who skateboard, 22% have never had an injury, 45% have had one injury, 18% have had two injuries, and 15% have had three or more injuries. What is the probability that a randomly chosen skateboarder has not had three or more injuries? 0.15 0.25 0.50 0.85

d

Executives for a car dealership are interested in the sales for the type of vehicle, SUV or truck, and the type of power train, two-wheel drive (2WD), four-wheel drive (4WD), or all-wheel drive (AWD). The data from the sales of 165 vehicles are displayed in the two-way table. A vehicle is randomly selected. Let S be the event that the vehicle is an SUV and let D be the event that the vehicle has 4WD. What is the value of P(S and DC)? 0.04 0.23 0.38 0.41

not a

A survey of 500 college students moving into their dorm revealed that 425 brought a microwave, 380 brought a video game console, and 50 brought neither a microwave nor a game console. A survey participant is randomly selected. Let M be the event that the participant brought a microwave and let C be the event that the participant brought a video game console. Organize these events in a two-way table. What is the probability that the participant brought both a microwave and a console, P(M and C)? 0.65 0.71 0.76 0.90

not c

The manufacturer of a soccer ball claims that only 3% of the soccer balls produced are faulty. An employee of this company examines the long-run relative frequency of faulty soccer balls produced as shown in the graph. Which conclusion can be drawn from this graph? The company's claim seems to be true because the graph shows that when 50 soccer balls were tested, only about 3% of them were faulty. We should not believe the company's claim that only 3% of their soccer balls are faulty because this graph shows a continuous increase in probability. Because the graph shows that the probability of producing a faulty soccer ball is 0.03, we can believe the company's claim that only 3% of the produced soccer balls are faulty The graph shows that the probability of producing a faulty soccer ball is about 0.06; therefore, we should not believe the company's claim that only 3% of the produced soccer balls are faulty.

not c

A study reports that about half of pet owners have either a dog or a cat. The other half of pet owners have birds, reptiles, or other animals. To find out if this applies in its area, an animal shelter surveys a random sample of 30 pet owners. Nineteen of them say they own either a cat or a dog. Let even digits represent owning a cat or a dog and odd digits represent owning some other type of animal. Using the line of random numbers, what is the best estimate of the proportion of responses in a sample who own either a cat or a dog? 0.47 0.50 0.53 0.63

not d probably b.

Dropping a piece of buttered toast will theoretically land butter-side down with a probability of 0.65. Some students decided to test this theory and dropped five pieces of buttered toast. All five landed butter-side down. One of the students claims that the next piece of buttered toast dropped will land butter-side up because it is due to happen. Is the student's reasoning correct? Yes, the students need to start dropping buttered toast to land butter up to get back to 0.65. No, the probability of buttered toast landing butter-side down is 0.65 over a large number of trials. No, if buttered toast lands butter-side down five times in a row, the probability must be higher than 0.65. Yes, it is unlikely that five pieces of dropped buttered toast will land butter-side down, so butter up must be due.

not d probably b.