Probability Questions and Answers

point s possible The data on the right represent the number of live multiple delivery births three or more babies in a particular year for women 15 to 54 years old Use the data to complete parts a through d below Age 15 19 20 24 25 29 30 34 35 39 40 44 45 54 Number of Multiple Births a Determine the probability that a randomly selected multiple birth for women 15 54 years old involved a mother 30 to 39 years old P 30 to 39 Type an integer or decimal rounded to three decimal places as needed 92 505 1630 2832 1847 379 117 b Determine the probability that a randomly selected multiple birth for women 15 54 years old involved a mother who was not 30 to 39 years old P not 30 to 39 Type an integer or decimal rounded to three decimal places as needed c Determine the probability that a randomly selected multiple birth for women 15 54 years old involved a mother who was less than 45 years old P less than 45 Type an integer or decimal rounded to three decimal places as needed d Determine the probability that a randomly selected multiple birth for women 15 5 years old involved a mother who was at least 40 years old Interpret this result Is it unusual Find the probability that a randomly selected multiple birth for women 15 54 years involved a mother who was at least 40 years old P at least 40 Type an integer or decimal rounded to three decimal places as needed

Use the order of operations to simplfy each expression Part 1 62 Part 2 6 Part 3 6

Compute the exact values 1 4

Evaluate 10

A survey of 600 randomly selected high school students determined that 536 play organized sports a What is the probability that a randomly selected high school student plays organized sports b Interpret this probability a The probability that a randomly selected high school student plays organized sports is Round to the nearest thousandth as needed b Choose the correct answer below Type a whole number OA If 1 000 high school students were sampled it would be expected that about of them play organized sports OB If 1 000 high school students were sampled it would be expected that exactly of them play organized sports

Evaluate 53

Write 7 7 7 7 7 7 in exponential notation Ilse to create an oxponent

a Find the probability that all three have type B blood The probability that all three have type B blood is Round to six decimal places as needed b Find the probability that none of the three have type B blood The probability that none of the three have type B blood is Round to three decimal places as needed c Find the probability that at least one of the three has type B blood The probability that at least one of the three has type B blood is Round to three decimal places as needed d Which of the events can be considered unusual Explain Select all that apply OA The event in part c is unusual because its probability is less than or equal to 0 05 OB None of these events are unusual OC The event in part b is unusual because its probability is less than or equal to 0 05 OD The event in part a is unusual because its probability is less than or equal to 0 05 7 of 40 people in the U CI

In a sample of 1200 U S adults 211 dine out at a resaurant more than once per week Two U S adults are selected at random from the population of all U S a without replacement Assuming the sample is representative of all U S adults complete parts a through d a Find the probability that both adults dine out more than once per week The probability that both adults dine out more than once per week is Round to three decimal places as needed b Find the probability that neither adult dines out more than once per week The probability that neither adult dines out more than once per week is Round to three decimal places as needed c Find the probability that at least one of the two adults dines out more than once per week The probability that at least one of the two adults dines out more than once per week is Round to three decimal places as needed d Which of the events can be considered unusual Explain Select all that apply A The event in part a is unusual because its probability is less than or equal to 0 05 B The event in part b is unusual because its probability is less than or equal to 0 05 C The event in part c is unusual because its probability is less than or equal to 0 05 D None of these events are unusual

Two cards are selected from a standard deck of 52 playing cards The first is replaced before the second card is selected Find the probability of selecting a heart and then selecting a nine The probability of selecting a heart and then selecting a nine is Round to three decimal places as needed 4

4 The table below shows the results of a survey in which 143 men and 145 women workers ages 25 to 64 were asked if they have at least one month s income set aside for emergencies Complete parts a through d Men Women Total 66 149 83 62 139 One month s income or more 77 Total 143 145 288 0 Less than one month s income a Find the probability that a randomly selected worker has one month s income or more set aside for emergencies The probability is Round to the nearest thousandth as needed b Given that a randomly selected worker is a male find the probability that the worker has less than one month s income The probability is Round to the nearest thousandth as needed c Given that a randomly selected worker has one month s income or more find the probability that the worker is a female The probability is Round to the nearest thousandth as needed d Are the events having less than one month s income saved and being male independent or dependent O Independent O Dependent

3 The table below shows the number of male and female students enrolled in nursing at a particular university for a recent semester a Find the probability that a randomly selected student is male given that the student is a nursing major b Find the probability that a randomly selected student is a nursing major given that the student is male Nursing Majors Non nursing majors Total 1148 2408 3556 C Males Females Total 92 733 825 1056 1675 2731 a Find the probability that a randomly selected student is male given that the student is a nursing major The probability is Round to three decimal places as needed b Find the probability that a randomly selected student is a nursing major given that the student is male The probability is Round to three decimal places as needed

c Does 30 minutes of aerobic exercise each day provide significant improvement in mental performance To investigate this issue a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months At the end of the study 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults Nine of these variables were significantly better in the sense of statistical significance at the a 0 05 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole and one variable was significantly better at the a 0 01 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole It would be correct to conclude O A that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the a 0 01 level We should be somewhat cautious about making claims for the variables that were significant at the a 0 05 level OB that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance C that these results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured OD none of the above

Problem 2 The error involved in making a certain measurement is a continuous rv X with the following pdf 0 0 f x x 2 x 2 otherwise 0 09375 4 x 0 Compute P X 0 5 or X 0 5 Round your answer to four decimal places

Let X be the total medical expenses in 1000s of dollars incurred by a particular individual during a given year Although X is a discrete random variable suppose its distribution is quite well approximated by a continuous distribution with pdf f x k 1 x 2 5 7 for x 0 a What is the value of k b Graph the pdf of X c What are the expected value and standard deviation of total medical expenses d This individual is covered by an insurance plan that entails a 500 deductible provision so the first 500 worth of expenses are paid by the individual Then the plan will pay 80 of any additional expenses exceeding 500 and the maximum payment by the individual including the deductible amount is 2500 Let Y denote the amount of this individual s medical expenses paid by the insurance company What is the expected value of Y Hint First figure out what value of X corresponds to the maximum out of pocket expense of 2500 Then write an expression for Y as a function of X which involves several different pieces and calculate the expected value of this function

81 Sales delay is the elapsed time between the manufacture of a product and its sale According to the article Warranty Claims Data Analysis Considering Sales Delay Quality and Reliability Engr Intl 2013 113 123 it is quite common for investigators to model sales delay using a lognormal distribution For a particu lar product the cited article proposes this distribution with parameter values 2 05 and 06 here the unit for delay is months a What are the variance and standard deviation of delay time b What is the probability that delay time exceeds 12 months C d e What is the probability that delay time is within one standard deviation of its mean value What is the median of the delay time distribution What is the 99th percentile of the delay time distribution f Among 10 randomly selected such items how many would you expect to have a delay time exceeding 8 months

A confidence interval for a population mean has a margin of error of 3 2 a Determine the length of the confidence interval b If the sample mean is 57 7 obtain the confidence interval Confidence interval

c According to the Merck Veterinary Manual the average resting heart rate for a certain type of sheep dog is 115 beats per minute bpm A Montana farmer notices his aging sheep dog has been acting more lethargic than usual and wonders if her heart rate is slowing He measures her heart rate on 15 occasions and finds a sample mean heart rate of 118 2 bpm a A Ho 115 H 115 B Ho p 118 2 Ha 118 2 C Ho 115 H 115 O D Ho 115 H 115 E Ho 118 2 Ha OF Ho 115 Ha 118 2 115

For each statement select the correct null hypothesis Ho and alternative hypothesis Ha in symbolic form a A certain type of hummingbird is known to have an average weight of 4 55 grams A researcher wonders if hummingbirds of this same type living in the Grand Canyon differ in weight from the population as a whole The researcher finds a sample of 30 such hummingbirds from the Grand Canyon and calculates their average weight to be 3 75 grams A Ho 3 75 B Ho 4 55 c Ho 4 55 D Ho 4 55 E Ho 3 75 F Ho 4 55 Ho 4 55 Ha 3 75 Ha 4 55 Ho p 4 55 H 4 55 H 3 75

Be sure to jully justify your responses when appropriate Show where all your numbers came from if not explicitly written Don t forget Units and use complete sentences when appropriate asked for 1 10 pts We bought a pack of 10 lightbulbs from a garage sale and 6 of them don t work If we were to collect a random sample of 2 lightbulbs no replacement and count the number of lightbulbs that are defective create a probability distribution for this discrete random variable a First define your random variable X The random variable X will be defined as the number of defective lightbulbs in a random sample of 2 lightbulbs chosen without replacement from the pack of 10 bulbs X can take from values 0 to 2 where X 0 means both bulbs are working X 1 means one bulbs is working and X 2 means both bulbs are defective b Next showing work for all of your probabilities in the space below fill in the table below round probabilities to 4 decimal places X P X

9 pts According to a study 67 of cars are some shade of grey white black grey or silver If we take a random sample of 25 cars find the following probabilities a None of them are a shade of gray

2 8 pts Scott has 8 class periods per week If we let our random variable C the number of classes Scott is on time to in a week the corresponding probabilities are included in the table below Use the table to answer the following questions 0 0 02 1 0 08 2 0 14 3 0 16 4 0 22 5 0 18 6 0 10 8 0 04 C 0 06 a In a random week what is the probability that Scott is on time to exactly 5 classes

You are measuring the concentration of stress hormone cortisol in saliva samples from volunteers and want to determine whether those who are parents have higher cortisol levels than non parents Which of these if true would reduce the statistical power of your study You can choose more than one Parents tend to be older and older people are less stressed There is no difference in stress levels between parents and nonparents You have measured only a very small number of parents You have measured only a very small number of non parents Parents tend to be older and older people are more stressed

Your test gives you a p value of 0 03 so you reject the null hypothesis that parents and non parents have the same mean of cortisol levels But what does your p value mean O There is still a 3 chance that the null hypothesis is correct O 97 of parents are more stressed than any of the non parents and vice versa If you repeated this study with a new set of samples from each population you would find a difference like this between parents and non parents 97 of the time You would find a difference like this between parents and non parents 3 of the time even if the two groups have the same cortisol levels There is only a 0 03 chance that the null hypothesis is correct

Dr Smith is trying to detect whether there is a correlation between soil acidity and the number of flowers on plants She grows a treatment group of plants on acidified soil and a control group on normal soil She measures 40 plants in each group and finds that there is a weak correlation but it is not quite significant p 0 054 so she measures another 10 plants in each group and recalculates the p value a 0 046 She rejects the null hypothesis In her paper she writes only that the sample size was 50 and the correlation was significant What is the problem with this statistical analysis O Cheating O Undisclosed flexibility O Confounding variables O Non independent replicates O Multiple comparisons

Suppose a random variable x arises from a binomial experiment If n 14 and p 0 2 find the following probabilities using the binomial formula Round to four decimal places if necessary P x 1 0 1539 P x 7 0 0092 P x 6 0 0322 P x 11 1 0000 P x 9 P x 4

According to the American Red Cross 8 1 of all Connecticut residents have Type B blood A random sample of 17 Connecticut residents is taken X the number of CT residents that have Type B blood of the 17 sampled What is the expected value of the random variable X 1 564 1 717 1 377 1 411 1 649 1 819 O O O

An insurance company prices its Tornado Insurance using the following assumptions In any calendar year there can be at most one tornado In any calendar year the probability of a tornado is 0 02 The number of tornadoes in any calendar year is independent of the number of tornados in any other calendar year Using the insurance company s assumptions calculate the probability that there are fewer than 2 tornadoes in a 12 year period Round your answer to four decimal places

A binomial experiment consists of 17 trials The probability of success on trial 10 is 0 42 What is the probability of success on trial 14 00 3 0 42 0 19 0 87 00 26 0 43

6 5 In an effort to link cold environments with hypertension in humans a preliminary experi ment was conducted to investigate the effect of cold on hypertension in rats Two random samples of 6 rats each were exposed to different environments One sample of rats was held in a normal environment at 26 C The other sample was held in a cold 5 C environment Blood pressures and heart rates were measured for rats for both groups The blood pressures for the 12 rats are shown in the accompanying table Rat 1 34563 a Do the data provide sufficient evidence that rats exposed to a 5 C environment have a higher mean blood pressure than rats exposed to a 26 C environment Use a 05 b Evaluate the three conditions required for the test used in part a c Provide a 95 confidence interval on the difference in the two population means 26 C Blood Pressure 152 157 179 182 176 149 Rat 789062 10 11 12 5 C Blood Pressure 384 369 354 375 366 423

126 Approximately 8 percent of students at a local high school participate in after school sports all four years of high school A group of 60 seniors is randomly chosen Of interest is the number who participated in after school sports all four years of high school a In words define the random variable X b List the values that X may take on c Give the distribution of X X d How many seniors are expected to have participated in after school sports all four years of high school e Based on numerical values would you be surprised if none of the seniors participated in after school sports all four years of high school Justify your answer numerically f Based on numerical values is it more likely that four or that five of the seniors participated in after school sports all four years of high school Justify your answer numerically

ble 116 A bridge hand is defined as 13 cards selected at random and without replacement from a deck of 52 cards In a standard deck of cards there are 13 cards from each suit hearts spades clubs and diamonds What is the probability of being dealt a hand that does not contain a heart a What is the group of interest b How many are in the group of interest c How many are in the other group d Let X What values does X take on e The probability question is PL f Find the probability in question g Find the 1 mean and ii standard deviation of X

4 6 Poisson 117 The switchboard in a Minneapolis law office gets an average of 5 5 incoming phone calls during the noon hour on Mondays Experience shows that the existing staff can handle up to six calls in an hour Let X the number of calls received at noon a Find the mean and standard deviation of X b What is the probability that the office receives at most six calls at noon on Monday c Find the probability that the law office receives six calls at noon What does this mean to the law office staff who get on average 5 5 incoming phone calls at noon d What is the probability that the office receives more than eight calls at noon with an average of 60

f Find the probability that at most three on the committee are not 115 Suppose that nine Massachusetts athletes are scheduled to appear at a charity benefit The nine are randomly chosen from eight volunteers from the local basketball team and four volunteers from the local football team We are interested in the number of football players picked a In words define the random variable X b List the values that X may take on c Give the distribution of X X d Are you choosing the nine athletes with or without replacement lected at random and without replacement from a deck of 52

able 93 A student takes a 32 question multiple choice exam but did not study and randomly guesses each answer Each question has three possible choices for the answer Find the probability that the student guesses more than 75 percent of the questions correctly

1 000 babies in an intensive care nursery Suppose that 1 000 babies from healthy baby nurseries were randomly surveyed Find the probability that exactly two babies were born deaf Use the following information to answer the next four exercises Recently a nurse commented that when a patient calls the medical advice line claiming to have the flu the chance that he or she truly has the flu and not just a nasty cold is only about 4 percent Of the next 25 patients calling in claiming to have the flu we are interested in how many actually have the flu 83 Define the random variable and list its possible values 84 State the distribution of X 85 Find the probability that at least four of the 25 patients actually have the flu 36 On average for every 25 patients calling in how many do you expect to have the flu 87 People visiting video rental stores often rent more than one DVD at a time The probability distribution for DVD rentals per customer at Video to Go is given Table 4 37 There is a five video limit per customer at this store so nobody ever rents more than five DVDs X 0 1 2 3 4 5 Table 4 37 P x 03 50 24 07 04 a Describe the random variable X in words b Find the probability that a customer rents three DVDs

ble Use the following information to answer the next two exercises The average number of times per week that Mrs Plum s cats wake her up at night because they want to play is 10 We are interested in the number of times her cats wake her up each week 128 In words what is the random variable X a the number of times Mrs Plum s cats wake her up each week b the number of times Mrs Plum s cats wake her up each hour c the number of times Mrs Plum s cats wake her up each night d the number of times Mrs Plum s cats wake her up 2 129 Find the probability that her cats will wake her up no more than five times next week a 5000 b 9329 c 0378 d 0671

118 The maternity ward at a hospital in the Philippines is one of the busiest in the world with an average of 60 births per day Let X the number of births in an hour a Find the mean and standard deviation of X b Sketch a graph of the probability distribution of X c What is the probability that the maternity ward will deliver three babies in one hour d What is the probability that the maternity ward will deliver at most three babies in one hour e What is the probability that the maternity ward will deliver more than five babies in one hour up that 3 percent of its bulbs are defective Using both the

109 Ellen has music practice three days a week She practices for all of the three days 85 percent of the time two days 8 percent of the time one day 4 percent of the time and no days 3 percent of the time One week is selected at random What values does X take on

111 According to a recent poll 75 percent of millennials people born between 1981 and 1995 have a profile on a social networking site Let X the number of millennials you ask until you find a person without a profile on a social networking site a Describe the distribution of X b Find the 1 mean and ii standard deviation of X c What is the probability that you must ask 10 people to find one person without a social networking site d What is the probability that you must ask 20 people to find one person without a social networking site phobility that you must ask at most five people

orten rent more than one DVD at a time The probability distribution for DVD rentals per customer at Video to Go is given in the following table There is a five video limit per customer at this store so nobody ever rents more than five DVDs 0 1 2 3 4 5 Table 4 35 X 0 1 2 a Describe the random variable X in words b Find the probability that a customer rents three DVDs c Find the probability that a customer rents at least four DVDs 3 d Find the probability that a customer rents at most two DVDs Another shop Entertainment Headquarters rents DVDs and video games The probability distribution for DVD rentals per customer at this shop is given as follows They also have a five DVD limit per customer 4 P x 5 03 Table 4 36 50 24 70 04 P x 35 25 20 10 05 05 V e At which store is the expected number of DVDs rented per customer higher f If Video to Go estimates that they will have 300 customers next week how many DVDs do they expect to rent next week Answer in sentence form g If Video to Go expects 300 customers next week and Entertainment Headquarters projects that they will have 420 customers for which store is the expected number of DVD rentals for next week higher Explain h Which of the two video stores experiences more variation in the number of DVD rentals per customer How do you know that

Use the following information to answer the next three exercises The probability that a local hockey team will win any given game is 0 3694 based on a 13 year win history of 382 wins out of 1 034 games played as of a certain date An upcoming monthly schedule contains 12 games 89 What is the expected number of wins for that upcoming month a 1 67 b 12 382 C 1043 d 4 43

79 A friend offers you the following deal For a 10 fee you may pick an envelope from a box containing 100 seemingly identical envelopes However each envelope contains a coupon for a free gift Ten of the coupons are for a free gift worth 6 Eighty of the coupons are for a free gift worth 8 Six of the coupons are for a free gift worth 12 Four of the coupons are for a free gift worth 40 Based upon the financial gain or loss over the long run should you play the game a Yes I expect to come out ahead in money b No I expect to come out behind in money c It doesn t matter I expect to break even cileble

a About what percent of the cheese last between 10 and 15 days P 10 x 15 21 10 15 5 1 6 3 P 10 x415 0 0548 0 9452 F 94 527 b About what percent of the cheese last between 13 and 16 days D 13 x 14 2 1 13 10 13 16 3 3 c The oldest 10 of cheese are put on sale These cheeses will be older than what age d From a random selection of 60 cheeses how many would you expect to be over a week old

The Blacktop Speedway is a supplier of automotive parts Included in stock are 12 speedometers that are correctly calibrated and two that are not Three speedometers are randomly selected without replaceme Let the random variable a represent the number that are not correctly calibrated Complete the probability distribution table Report probabilities accurate to 4 decimal places P x X 0 1

Keke s Kookies https kekekookies com sells mini cookies in packs of 5 and has determined a probability distribution for the number of cookies that they sell in a given day down X sold 0 5 10 15 20 Probability 0 21 0 12 0 37 0 11 a What is the probability of selling 15 mini cookies in a given day b Find the expected number of mini cookies sold in a day using the discrete probability distribution

Find the mean of the following probability distribution Round your answer to one decimal X P x 0 0 0947 1 0 0965 2 0 1102 3 0 4188 4 0 2799

The following table represents the probability of the number of cars owned by a college student Give your answer to at least 3 decimal places X 0 1 2 3 P x 0 5 0 35 0 12 0 03 a Is this a valid discrete probability distribution Select an answer b Find the mean number of cars owned c Find the standard deviation of the number of ca d Find 2 Question Help Message instructor Submit Question Select an answer No since P x 1 and 0 P x 1 Yes since P x 1 and 0 P x 1 Yes since P x 1 and P x 0

The probability that a randomly selected 4 year old male garter snake will live to be 5 years old is 0 95842 a What is the probability that two randomly selected 4 year old male garter snakes will live to be 5 years old b What is the probability that five randomly selected 4 year old male garter snakes will live to be 5 years old c What is the probability that at least one of five randomly selected 4 year old male garter snakes will not live to be 5 years old Would it be unusual if at least one of five randomly selected 4 year old male garter snakes did not live to be 5 years old a The probability that two randomly selected 4 year old male garter snakes will live to be 5 years old is Round to five decimal places as needed b The probability that five randomly selected 4 year old male garter snakes will live to be 5 years old is Round to five decimal places as needed c The probability that at least one of five randomly selected 4 year old male garter snakes will not live to be 5 years old is Round to five decimal places as needed 0 05 Would it be unusual if at least one of five randomly selected 4 year old male garter snakes did not live to be 5 years old because the probability of this happening is

Suppose that events E and F are independent P E 0 6 and P F 0 9 What is the P E and F The probability P E and F is intener or a decimal