BUSI 230 Review Exam 2 Probability solutions complete answers

BUSI 230 Review Exam 2 Probability solutions complete answers 

 

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Standardized Test: You are trying to answer a multiple choice question on a standardized test. There are four choices. If you get the question right, you gain one point, and if you get it wrong, you lose  point. Assume you have no idea what the right answer is, so you pick one of the choices at random. What is the expected value of the number of points you get?

 

GED: In a certain high school, the probability that a student drops out is 0.06, and the probability that a dropout gets a high-school equivalency diploma (GED) is 0.28. What is the probability that a randomly selected student gets a GED? Round your answer to four decimal places, if necessary.

The probability that the randomly selected student gets a GED is

 

Pay Your bills: A company audit showed that of 962 bills that were sent out, 629 were paid on time, 102 were paid up to 30 days late, 65 were paid between 31 and 90 days late, and 166 remained unpaid after 90 days. One bill is selected at random.

(a) What is the probability that the bill was paid on time? Round your answer to four decimal places.

(b) What is the probability that the bill was not paid on time? Round your answer to four decimal places.

 

Hospital visits: According to a health agency, there were  hosital visits for asthma–related illnesses in a recent year. The age distribution was as follows. Round your answers to four decimal places if necessary.

(a) What is the probability that an asthma patient is between 18 and 44 years old?

(b) What is the probability than an asthma patient is less than 1 year old?

(c) Using a cutoff of 0.05, is it unusual for an asthma patient to be less than 85 years old or older?

 

Determine whether events A and B are mutually exclusive.

A: Stacey is pursuing a major in biochemistry.
B: Stacey is pursuing a minor in animal sciences.
These events
 
A: Kim lives in Michigan.
B: Kim lives in Ohio.
These events
 
A red die and a blue die are rolled.
A: The red die comes up 5.
B: The total is 8.
These events
 
A: Sophie was valedictorian for her high school graduating class.
B: Sophie was salutatorian for her high school graduating class.
These events
 
A: Spencer has a part-time job at Starbucks.
B: Spencer attends college full time.
These events
 
A: Benson earned an A in his history class this term.
B: Benson earned a C in his history class this term.
These events
 
A: Benson earned an A in his history class this term.
B: Benson earned a C in his history class this term.
These events
 
A sample of 25 students is selected from the list of all students in a large statistics class.
A: At least 4 of the students are majoring in psychology.
B: Fewer than 4 of the students are majoring in psychology.
These events
 
A: Luis has a dog.
B: Luis has a cat.
These events
 
A: Jayden has a math class on Tuesdays at 2:00
B: Jayden has an English class on Wednesdays at 2:00
These events
 
A sample of 30 books is selected from the library.
At least 10 of the books are mysteries.
At least 10 of the authors are male.
These events
 

A sample of 100 cell phone batteries was selected. Find the complements of the following events.

(a)More than 138 of them use Google as their primary search engine. 

(b)At least  of them use Google as their primary search engine.

(c)Fewer than  of them use Google as their primary search engine.

(d)Exactly  of them use Google as their primary search engine.

 

How are your grades? In a recent semester at a local university, 600 students enrolled in both Statistics I and Psychology I. Of these students, 90 got an A in statistics, 79 got an A in psychology, and 31 got an A in both statistics and psychology. Round the answers to four decimal places, as needed.

(a) Find the probability that a randomly chosen student got an A in statistics or psychology or both.

(b) Find the probability that a randomly chosen student did not get an A in psychology.

 

Paving stones:  224 paving stones were examined for cracks, and 16 were found to be cracked. The same 224 stones were examined for discoloration, and 25 were found to be discolored. A total of 11 stones were both cracked and discolored. One of the 224 stones is selected at random. Round all answers to four decimal places.

(a)Find the probability that it is cracked.

(b)Find the probability that it is discolored.

(c)Find the probability that it is not cracked.

(d)Find the probability that it is cracked or discolored.

 

Let A and B be events with P(A)=0.6, P(B)=0.4, and P(A and B)=0.02.

Are A and B independent? Explain.

 

Genetics: A geneticist is studying two genes. Each gene can be either dominant or recessive. A sample of 100 individuals is categorized as follows. Write your answer as a fraction or a decimal, rounded to four decimal places.

(a) What is the probability that in a randomly sampled individual, gene 1 is recessive?

(b) What is the probability that in a randomly sampled individual, gene 2 is recessive?

(c) Given that gene 1 is recessive, what is the probability that gene 2 is recessive?

(d) Two genes are said to be in linkage equilibrium if the event that gene 1 is recessive is independent of the event that gene 2 is recessive. Are these genes in linkage equilibrium?

 

(a) What is the probability that in a randomly sampled individual, gene 1 is dominant?

(b) What is the probability that in a randomly sampled individual, gene 2 is dominant?

(c) Given that gene 1 is dominant, what is the probability that gene 2 is dominant?

(d) Two genes are said to be in linkage equilibrium if the event that gene 1 is dominant is independent of the event that gene 2 is dominant. Are these genes in linkage equilibrium?

 

Do you know Squidward? According to a survey by Nickelodeon TV, 88% of children under 13 in Germany recognized a picture of the cartoon character SpongeBob SquarePants. Assume that among those children, 70% also recognized SpongeBob's cranky neighbor Squidward Tentacles. What is the probability that a German child recognized both SpongeBob and Squidward? Write your answer as a fraction or a decimal, rounded to four decimal places.

The probability that the German child recognized both SpongeBob and Squidward is

 

The probability that a certain make of car will need repairs in the first six months is 0.1. A dealer sells four such cars. What is the probability that at least one of them will require repairs in the first six months? Round your final answer to four decimal places.

P(At least one car will require repairs)=

 

Evaluate the permutation.

Evaluate the combination.

 

Playing the horses: In horse racing, one can make a trifecta bet by specifying which horse will come in first, which will come in second, and which will come in third, in the correct order. One can make a box trifecta bet by specifying which three horses will come in first, second, and third, without specifying the order.

(a) In a six-horse field, how many different ways can one make a trifecta bet?

(b) In a six-horse field, how many different ways can one make a box trifecta bet?

 

Determine whether the random variable described is discrete or continuous.

The number of hospital admissions on a given day.

The number of credit cards owned by a randomly chosen person.

The weight of a block of tofu.

The amount of rain during the next thunderstorm.

The width of an oak tree.

The number of defective cell phones in a shipment of 50.

The weight of a randomly chosen student's backpack.

The temperature readings by a randomly selected city in the month of August.

The number of spades in a 5 card poker hand dealt from an ordinary deck of 52 cards.

The number of songs on a randomly chosen iPod.

The temperature readings by a randomly selected city in the month of August.

The amount of time of a randomly chosen college student to complete a statistical final exam.

The volume of an orange.

The number of heads in 100 tosses of a coin.

The height of a randomly chosen college student.

The number of babies born on January 1, 2011 at a randomly chosen hospital.

The number of attendees at a cinema on a Friday night. The weight of a randomly chosen college student.

The number of cars parked at a randomly chosen paid parking lot.

 

Dirty air: The federal government has enacted maximum allowable standards for air pollutants such as ozone. Let  be the number of days per year that the level of air pollution exceeds the standard in a certain city. The probability distribution of  is given by

(a) Compute the mean . Round the answer to at least two decimal places.

(b) Compute the standard deviation . Round the answer to at least three decimal places.

 

Standardized Test: You are trying to answer a multiple choice question on a standardized test. There are five choices. If you get the question right, you gain one point, and if you get it wrong, you lose 1/4 point. Assume you have no idea what the right answer is, so you pick one of the choices at random. What is the expected value of the number of points you get?

What is the expected value of the number of points you get? Round the answer to two decimal places.

 

Insurance: An insurance company sells a 1-year term life insurance policy to an 76-year-old woman. The woman pays a premium of $1,100. If she dies within 1 year, the company will pay $38,000 to her beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 76-year-old woman will be alive 1 year later is 0.9714. Let  be the profit made by the insurance company.

(a) Find the probability distribution.

(b) Find the expected value of the profit.

 

Determine the probability P(2) for a binomial experiment with n=4 trials and success probability p=0.8. Then find the mean, variance, and standard deviation.

Determine the probability . Round the answer to at least four decimal places.

Find the mean. Round the answer to two decimal places.

Find the variance and standard deviation. Round the variance to at least two decimal places and standard deviation to at least three decimal places.

 

Determine the probability P(Fewer than 2) for a binomial experiment with n=9 trials and success probability p=0.5. Then find the mean, variance, and standard deviation.

Determine the probability P(More than 13) for a binomial experiment with n=15 trials and success probability p=0.75. Then find the mean, variance, and standard deviation.

Determine the probability P(1 or fewer) for a binomial experiment with n=8 trials and success probability p=0.3. Then find the mean, variance, and standard deviation.

 

Google it: According to a recent report, 65% of Internet searches in a particular month used the Google search engine. Assume that a sample of 24 searches is studied. Round the answers to at least four decimal places.

(a)What is the probability that exactly 20 of them used Google?

(b)What is the probability that 15 or fewer used Google?

(c)What is the probability that more than 20 of them used Google?

(d)Would it be unusual if fewer than 13 used Google? Use a cutoff of 0.05.

 

College bound: A national college researcher reported that 65% of students who graduated from high school in  enrolled in college. Thirty two high school graduates are sampled. Round the answers to four decimal places.

(a)What is the probability that exactly 20 of them enroll in college?

(b)What is the probability that more than 15 enroll in college?

(c)What is the probability that fewer than 14 enroll in college?

(d)Would it be unusual if more than 23 of them enroll in college?

 

College bound: A national college researcher reported that 66% of students who graduated from high school in  enrolled in college. Twenty seven high school graduates are sampled.

(a) What is the mean number who enroll in college in a sample of  high school graduates? Round the answer to two decimal places.

(b) What is the standard deviation of the number who enroll in college in a sample of  high school graduates? Round the answer to four decimal places.

 

Find each of the shaded areas under the standard normal curve using a TI-84 calculator. Round the answers to four decimal places.

 

Use Excel to find the -score for which the area to its left is 0.52. Round the answer to two decimal places.

The z-score for the given area is

 

Use Excel to find the -score for which the area to its right is 0.61. Round the answer to two decimal places.

 

Use Excel to find the z-scores that bound the middle 68% of the area under the standard normal curve. Enter the answers in ascending order. Round the answers to two decimal places.

 

Big chickens: The weights of broilers (commercially raised chickens) are approximately normally distributed with mean 1348 grams and standard deviation 171 grams. Use the TI-84 Plus calculator to answer the following.

(a) What proportion of broilers weigh between 1100 and 1204 grams?

(b) What is the probability that a randomly selected broiler weighs more than 1510 grams?

(c) Is it unusual for a broiler to weigh more than 1555 grams?

 

Big chickens: A report from a poultry industry news Web site stated that the weights of broilers (commercially raised chickens) are approximately normally distributed with mean 1535 grams and standard deviation 169 grams. Use the TI-84 PLUS calculator and round your answers to at least two decimal places.

(a) Find the 23 percentile of the weights.

(b) Find the 96 percentile of the weights.

(c) Find the third quartile of the weights.

(d) A chicken farmer wants to provide a money-back guarantee that his broilers will weigh at least a certain amount. What weight should he guarantee so that he will have to give his customers' money back only 1% of the time?

 

Tree heights: Cherry trees in a certain orchard have heights that are normally distributed with mean u=112 inches and standard deviation =17 inches. Use the Excel spreadsheet to answer the following. Round the answers to at least four decimal places.

(a) What proportion of trees are more than 123 inches tall?

(b) What proportion of trees are less than 97 inches tall?

(c) What is the probability that a randomly chosen tree is between 95 and 110 inches tall?

 

Risky drivers: An automobile insurance company divides customers into three categories: good risks, medium risks, and poor risks. Assume that of a total of 11,078 customers, 7,734 are good risks, 2,401 are medium risks, and 943 are poor risks. As part of an audit, one customer is chosen at random. Round your answers to four decimal places if necessary.

 

Sick computers: Let V be the event that a computer contains a virus, and let W be the event that a computer contains a worm. Suppose P(V)=0.11, P(W)=0.28, P(V and W)=0.1.

 

Quality control: A population of  semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification, with the results shown in the following table. A wafer is chosen at random from the population. Write your answer as a fraction or a decimal, rounded to four decimal places.

(a) What is the probability that the wafer is from Lot B?

(b) What is the probability that the wafer is nonconforming?

(c) What is the probability that the wafer is from Lot B and is nonconforming?

(d) Given that the wafer is from Lot B, what is the probability that it is nonconforming?

(e) Given that the wafer is nonconforming, what is the probability that it is from Lot B?

(f) Let E1 be the event that the wafer comes from Lot B, and let E2 be the event that the wafer is nonconforming. Are E1 and E2 independent?

 

Books: Josephine has ten chemistry books, three history books, and nine statistics books. She wants to choose one book of each type to study. In how many ways can she choose the three books?

The total number of ways Josephine can choose the three books is

 

Ice cream: A certain ice cream parlor offers sixteen flavors of ice cream. You want an ice cream cone with three scoops of ice cream, all different flavors.

In how many ways can you choose a cone if it matters which flavor is on top, which is in the middle and which is on the bottom?

In how many ways can you choose a cone if the order of the flavors doesn't matter?

 

Fifteen items or less: The number of customers in line at a supermarket express checkout counter is a random variable with the following probability distribution.

 

Car inspection: Of all the registered automobiles in a city, 10% fail the emissions test. Ten automobiles are selected at random to undergo an emissions test. Round the answers to four decimal places.

(a) Find the probability that exactly three of them fail the test.

(b) Find the probability that fewer than three of them fail the test.

(c) Find the probability that more than two of them fail the test.

(d) Would it be unusual for more than one of them to fail the test?

 

Use the TI-84 Plus calculator to find the -score for which the area to its left is 0.59. Round the answer to two decimal places.

 

Use the TI-84 Plus calculator to find the -score for which the area to its right is 0.32. Round the answer to two decimal places.

 

Use the TI-84 Plus calculator to find the -scores that bound the middle 96% of the area under the standard normal curve. Enter the answers in ascending order and round to two decimal places.

 

Fish story: According to a report by the U.S. Fish and Wildlife Service, the mean length of six-year-old rainbow trout in the Arolik River in Alaska is 478 millimeters with a standard deviation of 38 millimeters. Assume these lengths are normally distributed.

(a) What proportion of six-year-old rainbow trout are less than 442 millimeters long?

(b) What proportion of six-year-old rainbow trout are between 400 and 500 millimeters long?

(c) Is it unusual for a six-year-old rainbow trout to be less than 372 millimeters long?

 

Fish story: According to a report by the U.S. Fish and Wildlife Service, the mean length of six-year-old rainbow trout in the Arolik River in Alaska is 481 millimeters with a standard deviation of 41 millimeters. Assume these lengths are normally distributed.

 

Electricity bills: According to a government energy agency, the mean monthly household electricity bill in the United States in  was $109.77. Assume the amounts are normally distributed with standard deviation $22.00. Use the TI-84 Plus calculator to answer the following.

(a) What proportion of bills are greater than 137?

(b) What proportion of bills are between $85 and $148?

(c) What is the probability that a randomly selected household had a monthly bill less than $119?

 

Electricity bills: According to a government energy agency, the mean monthly household electricity bill in the United States in  was $108.87. Assume the amounts are normally distributed with standard deviation $21.00. Use the TI-84 Plus calculator to answer the following.

 

New car: At a certain car dealership, the probability that a customer purchases an SUV is 0.16. Given that a customer purchases an SUV, the probability that it is black is 0.21. What is the probability that a customer purchases a black SUV? Round your answer to four decimal places, if necessary.

 

Books: Josephine has three chemistry books, three history books, and six statistics books. She wants to choose one book of each type to study. In how many ways can she choose the three books?

The total number of ways Josephine can choose the three books is

 

Do you carpool? Let X represent the number of occupants in a randomly chosen car on a certain stretch of highway during morning commute hours. A survey of cars showed that the probability distribution of X is as follows.

 

Business projection: An investor is considering a $10,000 investment in a start-up company. She estimates that she has probability 0.40 of a $21,000 loss, probability 0.24 of a $7,100 profit, probability 0.11 of a $53,000 profit, and probability 0.25 of breaking even (a profit of $0). What is the expected value of the profit? Would you advise the investor to make the investment?