Probability Questions and Answers

Suppose you just purchased a digital music player and have put 11 tracks on it After listening to them you decide that you like 2 of the songs With the random feature on your player each of the 11 songs is played once in random order Find the probability that among the first two songs played a You like both of them Would this be unusual b You like neither of them c You like exactly one of them d Redo a c if a song can be replayed before all 11 songs are played

Suppose the data below represent in thousands the type of health insurance coverage of people by age Determine P 18 years old and P 18 years old no health insurance Are the events 18 years old and no health insurance independent Age Private health insurance Government health insurance No health insurance Total 18 18 44 44 055 73 781 20 801 11 297 8 663 28 422 73 519 45 64 58 297 10 625 10 803 113 500 79 725 P 18 years old Round to three decimal places as needed 64 25 989 39 822 899 66 710 S Total 202 122 82 545 48 787 333 454

Suppose you just received a shipment of eight televisions Four of the televisions are defective If two televisions are andomly selected compute the probability that both televisions work What is the probability at least one of the two televisions does not work The probability that both televisions work is logos as needed

Due to a manufacturing error six cans of regular soda were accidentally filled with diet soda and placed into a 18 pack Suppose that two cans are randomly selected from the 18 pack Complete parts a through c a Determine the probability that both contain diet soda Round to four decimal places as needed with

Suppose that E and F are two events and that P E 0 6 and P FIE 0 2 What is P E and F P E and F

O 9 In its monthly report the local animal shelter states that it currently has 24 dogs and 18 cats available for adoption Eight of the dogs and 6 of the cats are male Find each of the following conditional probabilities if an animal is selected at random Hint create a two way table a Pipet is male cat b P pet is cat female c P pet is female dog 10 A private college report contains these statistics 70 of incoming freshmen attended public schools 75 of public school students who enroll as freshmen eventually graduate 90 of other freshmen eventually graduate Based on the probabilities given what percent of freshmen eventually graduate Hint organize the information in a tree diagram Answers 3 a 52 b S 888 BBG BGB BGG GBB GBG GGB GGG 1 7 6 a 8 b 8 6 9 2 a 36 b 36 b 52 12 a 2652 a 10 000 al combination 676 b 2652 c 8 c 36 16 c 52 24 c 132600 b 1 757 600 8 d 36 d c 2 433 600 6 20 20 e 36 e 52 d 916 132 832 c permutation d combination 7 a 20 1 720 8 a 0 06 b 10 120 d 120 g 120 b 0 50 0 94 0 429 0 333 9 a 18 10 79 5 12 b 28 16 c 24 e 13 983 87 0 667

Suppose that you are buying a car for 60 000 You saved 20 000 for a down payment The dealer is offering you two incentives Incentive A is a 5000 off the price of the car followed by a five year loan at 7 39 compounded monthly Incentive B does not have a cash rebate but provides free financing no interest over five years What is the difference between the monthly payments for the two offers Which incentive is the better deal P Payment nxt approximately 33 Incentive B is the better deal approximately 15 Incentive A is the better deal approximately 85 Incentive B is the better deal approximately 67 Incentive A is the better deal

A poll reported that 64 of adults were satisfied with the job the major airlines were doing Suppose 15 adults are selected at random and the number who are satisfied is recorded Complete parts a through e below a Explain why this is a binomial experiment Choose the correct answer below OA This is a binomial experiment because there are two mutually exclusive outcomes for each trial there is a fixed number of trials the outcome of one trial does not affect the outcome of another and the probability of success is the same for each trial B This is a binomial experiment because there are two mutually exclusive outcomes for each trial there is a fixed number of trials the outcome of one trial does not affect the outcome of another and the probability of success changes in each trial OC This is a binomial experiment because there are two mutually exclusive outcomes for each trial there is a random number of trials the outcome of one trial does not affect the outcome of another and the probability of success is the same for each trial OD This is a binomial experiment because there are three mutually exclusive outcomes for each trial there is a fixed number of trials the outcome of one trial does not affect the outcome of another and the probability of success is the same for each trial

Question 5 of 14 5 What is the second coordinate of the point at which the line y 0 5x 1 5 crosses the y axis Answer Previous Answer 1 5

8 How many units does the point 2 4 vertically deviate above the line y 2x 2 r MyHer Previous Answer Return to overview

Question 4 of 14 4 What is the slope of the line passing through the points 1 2 and 4 6 Answer Previous Answer 0 8

6 19 Student study times A class survey in a large class for first year college students asked About how many hours do you study during a typical week The mean response of the 463 students was as x 13 7 hours 6 Sup pose that we know that the study time follows a Normal distribution with standard deviation o 7 4 hours in mobo the population of all first year students at this university boird a Use the survey result to give a 99 confidence inter val for the mean study time of all first year students b What condition not yet mentioned must be met for onfidence interval to be valid

9 Sample Size and Margin of Error Example 16 1 page 377 described NHANES survey data on the body mass index BMI of 654 young women The mean BMI in the sample was x 26 8 We treated these data as an SRS from a Normally distributed population with standard deviation o 7 5 1 a Suppose that we had an SRS of just 100 young women What would be the margin of error for 95 confidence b Find the margins of error for 95 confidence based on SRSS of 400 young women and 1600 young women c Compare the three margins of error How does increasing the sample size change the margin of error of a confidence interval when the confidence level and population standard deviation remain the same

companying graph is a histogram of measured mid upper arm circumferences of 297 particular adults ick the icon to view the histogram stogram is point s possible symmetric The histogram is V St O Question 8 O Question 9 O Question 10 S The Frequency 50 40 30 20 10 0 20 Adult Mid Upper Arm Circumferences 24 36 40 28 32 Circumference centimeters 44 48

Opinion polls find that 20 of American adults claim that they don t get enough sleep Suppose you take a random sample of 200 American adults and count the number X in your sample that claim that they never have time to relax The standard deviation of X is O A 40 OB 6 32 O C 5 66 O D 32

A small class has 10 students Seven of the students are male and three are female You write the name of each student on a small card The cards are shuffled thoroughly and you choose one at random observe the name of the student and replace it in the set The cards are thoroughly reshuffled and you again choose a card at random observe the name and replace it in the set This is done a total of five times Let X be the number of cards observed in these four trials with a name corresponding to a male student The random variable X has which probability distribution O A the Normal distribution with mean 3 and variance 1 B the binomial distribution with parameters n 5 and p 0 3 C the binomial distribution with parameters n 5 and p 0 7 O D None of the above

The following table gives the sex and age group of college students at a midwestern university Female Male Total 89 5 668 1 904 1 660 9 321 15 to 17 years 18 to 24 years 25 to 34 years 35 years or older Total 61 4 697 1 589 970 7 317 150 10 365 3 493 2 630 16 638 A student is to be selected at random Given that the selected student is female the conditional probability that she is 25 to 34 years old is O A 0 008 OB 0 545 O C 0 114 O D 0 204

Event A occurs with probability 0 1 Event B occurs with probability 0 6 If A and B are independent then O A P A and B 0 70 B P A or B 0 64 OC P A or B 0 70 O D P A and B 0 64

A randomly selected sample of 100 horse owners found that 72 of them feed two flakes of grass hay in the morning and one flake of alfalfa plus one flake of grass hay in the evening to their horses while the rest feed two flakes of grass hay in the morning and one flake of alfalfa plus oat hay in the evening The estimated probability that horse owners feed grass hay in the A M and alfalfa plus oat hay in the P M is O A 0 75 OB 0 50 OC 0 28 O D 0 72

Consider the normal curve in the figure to the right which gives relative frequencies in a distribution of men s heights The distribution has a mean of 70 9 inches and a standard deviation of 2 6 inches 60 62 64 66 68 70 72 74 78 78 height inches a What is the total area under the curve The total area under the curve is Round to the nearest hundredth as needed b Estimate the relative frequency of values less than 68 Choose the correct answer below A 0 13 OB 0 52 OC 0 87 OD 1 00 c Estimate the relative frequency of values greater than 68 Choose the correct answer below OA 1 00 B 0 87 OC 0 13 OD 0 52 d Estimate the relative frequency of values between 68 and 71 Choose the correct answer below A 1 00 OB 0 13 OC 0 61 OD 0 39 e Estimate the relative frequency of values greater than 71 Choose the correct answer below A 0 48 B 1 00 OC 0 17 OD 0 52

A local bakary has determined a probability distribution for the number of cheesecakes that they sell in a given day X sold 0 5 10 15 20 Probability 0 19 0 21 0 13 0 06 What is the probability of selling 15 cheesecakes in a given day What is the probability of selling at least 10 cheesecakes

Which of the following options defines a random variable X that could be helpful in solving the given problem Note that you are not being asked to actually solve the problem A school site selection committee is to be chosen randomly from 16 men and 11 women If the committee consists of 20 members chosen randomly from the 27 what is the probability that exactly of them are men OX the probability of 4 women being chosen for the committee OX the probability of 4 men being chosen for the committee OX the number of people chosen for the committee OX the number of men chosen for the committee

Suppose you have an experiment where you flip a coin three times You then count the number of heads a State the random variable b 1 c I Select an answer rv X the probability that you flip heads rv X number of coins flipped rv X the number of heads flipped when you flip a coin three times rv X the number of heads flipped rv X flipping a coin e number of heads smallest to largest P X number of heads what shape would it be

5 29 Daily activity It appears that people who are mildly obese are less active than leaner people One study looked at the average number of minutes per day that people spend standing or walking Among mildly obese people the mean number of minutes of daily ac tivity standing or walking is approximately Normally distributed with mean 373 minutes and standard de viation 67 minutes The mean number of minutes of daily activity for lean people is approximately Normally distributed with mean 526 minutes and standard devia tion 107 minutes A researcher records the minutes of activity for an SRS of five mildly obese people and an SRS of five lean people a What is the probability that the mean number of minutes of daily activity of the five mildly obese people exceeds 420 minutes b What is the probability that the mean number of minutes of daily activity of the five lean people ex

Recent research suggests that 45 of residents from a certain region have a home phone 94 have a cell phone and 42 of people have both What is the probability that a resident from the region has a a home or cell phone b neither a home phone nor a cell phone c a cell phone but no home phone a The probability of a resident having a home or cell phone is Type an integer or decimal rounded to two decimal places as needed

5 4 pts Complete the following table for the experiment rolling 2 dice and adding their values to find the sum of your roll Determine all possible sums you could roll along with their frequency and relative frequency you may leave these as unreduced fractions Sum Frequency Relative frequency 2 1 36 3 2 36 4 3 3 36 S 4 4 36 6 S 36 7 6 6 36 8 S S 36 9 4 4 36 10 3 3 36 2 2 36 12 1 36 Are the sums all equally likely to be rolled What sum is the most likely to be rolled What sum is the least likely to be rolled

Points 0 of 1 Suppose that 24 of people have a dog 31 of people have a cat and 18 of people own both What is the probability that someone owns a dog or a cat The probability of a person having a dog or a cat is Tyne an integer or a decimal

Suppose that 25 of people have a dog 18 of people have a cat and 7 of people own both What is the probability that someone owns a dog or a c The probability of a person having a dog or a cat is Typo on integer or a docimal U OT 1

total of M Ms 12 1 7 pts There are 12 M M s in a bag 3 of them are blue 5 are red and 4 of them are green We are going to draw 2 of them from the bag one at a time without replacing them Hint it might help to draw a probability tree to help you with this question Calculate the following probabilities showing your work and giving your solutions as percentages with 2 decimal places Drawing a red M M and then a green M M The probability of drawing a red M M is 5 12 This leaves 11 M Ms left with 4 of them being green making the probability of drawing a green M M 4 11 P red on 1st draw P green on 2nd draw green ad 0 151515 0 151515 100 15 15 Drawing both a blue and red M M their order doesn t matter sample space 12C event space 3c 5c 4c Sc Sc c 3 5 1 5 3 1 15 15 30 drawing any 2 M M s of the same color 120 Sample space event space 3c 5c 4c Sc 3 10 6 19 3 1 1 10 1 1 1 1 6 2 event space sample space 30 1202 30 al 2 0 4545 event space co Sc 3c 4c sample space 12 294 0 2879 Drawing any 2 M M s that are NOT all the same Hint there s a hard way and an easy way to answer this question both equally correct

A consumer organization estimates that over a 1 year period 16 of cars will need to be repaired once 9 will need repairs twice and 1 will require three or more repairs If you own two cars what is the probability that a neither will need repair b both will need repair c at least one car will need repair a The probability that neither will need repair is Do not round b The probability that both will need repair is Do not round c The probability that at least one car will need repair is Do not round Submit quiz

4 pts Complete the following table for the experiment rolling 2 dice and adding their values to find the sum of your roll Determine all possible sums you could roll along with their frequency and relative frequency you may leave these as unreduced fractions Sum Frequency Relative frequency

Suppose that 28 of people have a dog 25 of people have a cat and 19 of people own both What is the probability that someone owns a dog or a cat The probability of a person having a dog or a cat is

35 Multiple choice tests Here is a simple probability model for multiple choice tests Suppose each student has probability p of correctly answering a question cho sen at random from a universe of possible questions A strong student has a higher p than a weak student Answers to different questions are independent a Stacey is a good student for whom p 0 75 Use the Normal approximation to find the probabil ity that Stacey scores between 70 and 80 on a 100 question test b If the test contains 250 questions what is the prob ability that Stacey will score between 70 and 80 You see that Stacey s score on the longer test is more

A survey conducted by the CDC found that 45 9 of U S households used landline phones in 2016 This raises concerns about the accuracy of certain surveys as they depend on random digit dialing to households via landlines Researchers pick eight U S households at random Complete parts a through c a What is the probability that all eight of them have a landline Round to five decimal places as needed

Your list of favorite songs contains ten rock songs eight hip hop songs and seven K pop songs a What is the probability that a randomly played song is a hip hop song b What is the probability that a randomly played song is not K pop a The probability that a randomly played song is a hip hop song is Type an integer or a decimal Do not round

color in the second generation of a certain cross of sweet peas should be red or white in a 3 1 ratio That is each plant has probability 3 4 of having red blossoms and the blossom colors of separate plants are independent blickwinkel Alamy a What is the probability that exactly three out of four of these plants have red blossoms b What is the mean number of red blossomed plants when 60 plants of this type are grown from seeds c What is the probability of obtaining at least 45 red from

Comment on the quotation below from a climatologist What I think is our best determination is it will be a colder than normal winter said Pamela Naber Knox a Wisconsin state climatologist I m basing that on a couple of different things First in look at the past few winters there has been a lack of really cold weather Even though we are not supposed to use the law of averages we are due Choose the correct answer below OA There is no Law of Averages She would be wrong to think they are due for a harsh winter OB The climatologist s prediction is probably wrong Since the past few winters have not had very cold weather it is more likely that the next winter will also not have very cold weather OC The climatologist is likely inaccurate Weather is too naturally unpredictable for forecasting to be reliable OD The climatologist is right in her assessment that she should not use the Law of Averages Her forecast for colder weather is likely accurate

The plastic arrow on a spinner for a child s game stops rotating to point at a color that will determine what happens next Are the given probability assignments possible Why or why not a Since 5 Since C Since d Since e Since the sum of the probabilities is less than 1 the sum of the probabilities is less than 1 at least one probability is not between 0 and 1 this probability assignment the sum of the probabilities is greater than 1 each probability is between 0 and 1 and the sum of the probabilities is 1 this probability assignment ability assignment ability assignment ability assignment is possible possible possible V possible possible a b c d e Red 1 0 25 0 4 0 4 0 5 Yellow Green 0 0 25 0 0 25 0 2 0 2 0 3 1 1 0 2 0 1 Blue 0 0 25 0 3 1 8 0 2

to review Examples 7 7 7 8 and 7 9 7 2 before attempting these problems Review Constants A lightweight vertical spring of force constant k has its lower end mounted on a table You compress the spring by a distance d place a block of mass m on top of the compressed spring and then release the block The spring launches the block upward and the block rises to a maximum height some distance above the now relaxed spring Find the speed of the block just as it loses contact with the spring Express your answer in terms of the variables m d and constants g k xa A O Xb 06 a x x X X IXI X 10n X med

Every morning my neighbor goes out walking I observe that 20 of the time she walks with her beagle 70 of the time she walks with her golden retriever and 30 of the time she walks alone What is the probability she walks both dogs at the same time Hint use the general addition rule Choose the correct answer below O A 20 B 30 OC 70 OD There is not enough information to answer the question

Suppose that 22 of people have a dog 27 of people have a cat and 15 of people own both What is the probability that someone owns a dog or a cat The probability of a person having a dog or a cat is

A casino claims that its roulette wheel is truly random What should that claim mean Choose the correct answer below OA The same number will not come up twice in a row OB The probabilities of all numbers will add to 1 OC Every number will occur before any numbers repeat O D Every number is equally likely to occur

In a dresser are ten blue shirts seven red shirts and eight black shirts a What is the probability of randomly selecting a red shirt b What is the probability that a randomly selected shirt is not black a The probability of randomly selecting a red shirt is Type an integer or a decimal Do not round b The probability that a randomly selected shirt is not black is Type an integer or a decimal Do not round

Suppose that 65 of families living in a certain country own a laptop and 22 own a desktop computer The Addition Rule might suggest then that 87 of families own either a laptop or a desktop computer What s wrong with that reasoning Choose the correct answer below CCIDE OA The sum of the probabilities of the two given events does not equal 1 so this is not a legitimate probability assignment OB If one family owns a laptop or a desktop computer it can influence another family to also own a laptop or a desktop computer The events are not independent so the Addition Rule does not apply OC The reasoning is correct Thus 87 of families own either a laptop or a desktop computer OD A family may own both a laptop and a desktop computer The events are not disjoint so the Addition Rule does not apply

33 Hyundai sales in 2015 Hyundai Motor America sold 761 710 vehicles in the United States in 2015 with the U S built Elantra leading sales with 241 706 cars sold The other top selling nameplates in 2015 were the Sonata with 213 303 sold the Santa Fe with 118 134 sold and the Tucson with 63 591 sold The company wants to undertake a survey of 2015 Hyundai buyers to ask them about satisfaction with their purchase a What proportion of the Hyundais sold in 2015 were Elantras b If they plan to survey a simple random sample of 1000 Hyundai buyers what is the expected number and standard deviation of the number of Elantra buyers in the sample c What is the probability they will get fewer than

aerosolized vaccine for measles was developed in Mexico and has been used on more than 4 million children since 1980 Aerosolized vaccines have the advantages of being able to be administered by people without clinical training and do not cause injection associated infections The percentage of children developing an immune response to measles after receiving the subcutaneous injection of the vaccine is 95 and for those receiving the aero solized vaccine it is 85 There are 20 children to be vaccinated for measles using the aerosolized vaccine in a small rural village in India We are going to count the num ber who devel im oped an mune response to measles after vaccination Dr P Marazzi Science Source a Explain why this is a binomial setting b What is the probability that at least one child does not develop an immune response to measles after receiving the aerosolized vaccine What would be the probability that at least one child does not develop an immune response to measles if all children were

11 College Admissions A small liberal arts college in Ohio would like to have an entering class of 450 students next year Past experience shows that about 37 of the students admitted will decide to attend The college is planning to admit 1175 students Suppose that students make their decisions independently and that the probability is 0 37 that a randomly chosen student will accept the offer of admission a What are the mean and standard deviation of the number of students who accept the admissions offer from this college b Use the Normal approximation to approximate probability that the college gets more students than it wants Be sure to check that you can safely use the approximation c Use software or an online binomial calculator to compute the exact prob ability that the college gets more students than it wants How good is the approximation in part b d To decrease the probability of getting more students than are wanted does the college need to increase or decrease the number of students it admits Using software or an online binomial calculator what is the larg est number of students that the college can admit if administrators want the exact probability of getting more students than they want to be no

tion in the Search box a Enter 5 choose 2 and click Search What does Google return b You see that Google calculates the binomial coefficient 5 choose 2 What are the values of the binomial coefficients for 500 choose 2 and 500 choose 100 We expect that there are more ways to choose 100 than to choose 2 but how many more may be a surprise That e 107 in Google s answer means a 1 followed by 107 zeros c Google also does binomial probabilities Enter 10 choose 1 0 11 0 89 9 to find P X 1 in Example 14 5 page 336 What is

3 Boxes of Tiles Boxes of 6 inch slate flooring tile contain 40 tiles per box The count X is the number of cracked tiles in a box You have noticed that most boxes contain no cracked tiles but if there are cracked tiles in a box then there are usually several

In each of Exercises 14 1 through 14 3 X is a count Does X have a binomial distribution Give your reasons in each case 14 1 Working Cell Numbers When an opinion poll selects cell phone numbers at random to dial the cell phone exchange is first selected and then random digits are added to form a complete telephone number see Example 8 5 page 212 When using this procedure to generate random cell phone numbers approxi mately 55 of the cell numbers generated correspond to working numbers You watch a pollster dial 15 cell numbers that have been selected in this manner X is the number that correspond to a working number