Probability Questions and Answers

In parts a and b identify whether the events are disjoint independent or neither events cannot be both disjoint and independent a You and a randomly selected student from your class both earn D s in this course Odisjoint independent neither b You and your class partner both earn D s in this course independent neither Odisjoint c If two events can occur at the same time they must be independent true false

Yvetia ordered two pizzas for her family Both pizzas are rectangular The larger pizza s width is equal to the length of the smaller pizza and the larger pizza s length is twice the width of the smaller pizza The sum of the perimeters of the two pizzas is 96 inches and the sum of the areas is 288 square inches Determine the dimensions of each pizza

A drug test is accurate 98 of the time If the test is given to 1800 people who have not taken drugs what is the probability that at most 38 will test positive Probability Give your answers to at least 3 decimal places

Assume that a procedure yields a binomial distribution with a trial repeated n 20 times Use either the binomial probability formula or technology to find the probability of k 9 successes given the probability p 46 of success on a single trial Report answer accurate to 4 decimal places P X k

5 and p 0 2 Next record the mean and Construct a Binomial probability distrubtion table using n standard deviation of the distribution below Where necessary round all numbers to four decimal places P x fl x

7 A commuter must pass through 5 traffic lights on her way to work and will have to stop at each one that is red She estimates the probability model for the number of red lights she hits as shown below Complete parts a a X of red 0 3 P X x 0 04 0 16 a How many red lights should she expect to hit each day The commuter should expect to hit Round to one decimal place as needed b What s the standard deviation 1 0 24 2 0 34 4 0 15 5 0 07 red lights each day D The standard deviation is Round the final answer to one decimal place as needed Round all intermediate values to three decimal places as needed

5 A basketball announcer suggests that a certain player is streaky That is the announcer believes that if the player makes a shot the player is more likely to make the next shot As evidence the announcer points to a recent game where the player took 30 shots and had a streak of 10 made shots in a row Is this convincing evidence of streaky shooting by the player Assume that this player makes 50 of the shots and that the results of a shot do not depend on previous shots a Describe how you would carry out a simulation to estimate the probability that a 50 shooter who takes 30 shots in a game would have a streak of 10 or more made shots Do not perform the simulation b The dotplot displays the results of 50 simulated games in which this player took 30 shots What conclusion would you draw about whether this player was streaky Explain uh Longest Streak in Simulated Game 10 6 A statistics class asked an SRS of 100 students at their school whether they regularly recycle or not In the sample 55 students said that they recycle Is this convincing evidence that more than half of the students at the school would say they regularly recycle The dotplot shows the results of taking 200 SRSS of 100 students from a population in which the true proportion who recycle is 0 50 38 40 42 44 46 45 50 52 54 56 58 Simulated proportion who say Yes as a percent a Explain why the sample result 55 out of 100 said Yes does not give convincing evidence that more than half of the school s students recycle b Suppose instead that 63 students in the class s sample had said Yes Explain why this result would give convincing evidence that a majority of the school s students recycle

A large tank of fish from a hatchery is being delivered to a lake The hatchery claims that the mean length of fish in the tank is 15 inches and the standard deviation is 2 inches A random sample of 39 fish is taken from the tank Let x be the mean sample length of these fish What is the probability that x is within 0 5 inch of the claimed population mean Round your answer to four decimal places USE SALT

Express Courier Service has found that the delivery time for packages is normally distributed with mean 14 hours and standard deviation 3 hours USE SALT a For a package selected at random what is the probability that it will be delivered in 18 hours or less Round your answer to four decimal places b What should be the guaranteed delivery time on all packages in order to be 95 sure that the package will be delivered before this time Hint Note that 5 of the packages will be delivered at a time the guaranteed time period Round your answer to one decimal place hr

Three circle red on white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at the Wind Mountain archaeological site At one excavation a sample of 170 potsherds indicated that 70 were of the three circle red on white pattern How many ceramic potsherds must be found and identified if we are to be 95 confident that the sample proportion p of such potsherds is within 6 6 of the population proportion o three circle red on white patterns found at this excavation site Round your answer up to the nearest whole number USE SALT

2 A husband a wife decide to have children until they have at least one child of each gender The couple had seven girls in a row Their doctor assured them that they were much more likely to have a boy for their next child after all those girls Explain why the doctor is wrong 3 According to The Book of Odds the probability that a randomly selected U S adult drinks coffee on a given day is 0 56 a Explain what this probability means b If a researcher surveys 100 U S adults at random on the same day will exactly 56 of them have consumed coffee that day Explain 4 A very good professional baseball player gets a hit about 35 of the time over an entire season After the player failed to hit safely in six straight at bats a TV commentator said He is due for a hit by the law of averages Explain why the commentator is wrong

Abigail Bobby Carlos DeAnna and Emily go to the bagel shop for lunch every Thursday Each time they randomly pick 2 o the group to pay for lunch by drawing names from a hat Find the probability that Carlos or DeAnna or both ends up paying for lunch OP Carlos or DeAnna 7 10 0 70 OP Carlos or DeAnna 6 10 0 60 OP Carlos or DeAnna 4 10 0 40 OP Carlos or DeAnna 3 10 0 30 OP Carlos or DeAnna 5 10 0 50

Canada has two official languages English and French Choose a Canadian at random and ask What is your mother tongue Here is a distribution of responses combining many separate languages from the broad Asia Pacific region 5 Language English French Probability 0 63 0 22 Asian Pacific Other 0 06 0 09 Is this a valid probability model Why or why not Yes because all the probabilities sum to 1 Yes because each probability is between 0 and I and all the probabilities sum to 1 O No because this is not a complete list of the languages spoken in Canada No because all the probabilities do not sum to 1 Yes because each probability is between 0 and 1

5 A teacher uses a program that randomly chooses a student in the class to volunteer to answer a question The teacher will stop choosing students once a student gets the right answer but if the teacher is given four wrong answers in a row he will stop and help the class himself He estimates that 79 of the students in the class know the correct answer Assume that students are chosen with replacement Complete parts a through c below a Create a probability model for the number of students the teacher will call on until he receives the correct answer or stops to help the class Number of students 1 2 3 4 Probability Type integers or decimals Do not round b Find the expected number of students called on The expected number of students called on is Round to two decimal places as needed c Find the expected number of wrong answers that will be offered The expected number of wrong answers offered is Round to two decimal places as needed

A report describes a survey of 500 licensed drivers Each driver in the sample was asked if they would prefer to keep complete control of the car while driving to use a partially self driving car that allowed partial driver control or to turn full control over to a driverless car Suppose that it is reasonable to regard this sample as a random sample of licensed drivers in the United States and that you want to use the data from this survey to decide if there is evidence that fewer than half of all licensed drivers in the United States prefer to keep complete control of the car while driving LUSE SALT a Describe the shape center and variability of the sampling distribution of p for random samples of size 500 if the null hypothesis Ho p 0 50 is true Round your standard deviation to four decimal places The shape of the sampling distribution is approximately normal The sampling distribution is centered at We would not 0 small or smaller is b Would you be surprised to observe a sample proportion as small as p 0 49 for a sample of size 500 if the null hypothesis Ho p 0 50 were true Explain why or why not Consider the event surprising if the probability that the event occurs when Ho p 0 5 is true is less than 0 05 Round your answer to four decimal places MY NOTES The standard deviation of the sampling distribution is We would small or smaller is be surprised to observe a sample proportion of p 0 49 for a sample of size 500 if the null hypothesis Ho P 0 5 is true The probability of a sample proportion this which is greater than the signficance level of 0 05 c Would you be surprised to observe a sample proportion as small as p 0 46 for a sample of size 500 if the null hypothesis Ho p 0 50 were true Explain why or why not Consider the event surprising if the probability that the event occurs when Ho p 0 5 is true is less than 0 05 Round your answer to four decimal places O be surprised to observe a sample proportion of p 0 46 for a sample of size 500 if the null hypothesis Ho P 0 5 is true The probability of a sample proportion this which is less than B the signficance level of 0 05 which is less than Since the probability of a sample proportion this small or smaller is fewer than 50 of licensed drivers prefer to keep complete control of the car when driving d The actual sample proportion observed in the study was p 0 44 Based on this sample proportion is there convincing evidence that fewer than 50 of licensed drivers prefer to keep complete control of the car when driving or is the sample proportion consistent with what you would expect to see when the null hypothesis is true Support your answer with a probability calculation Round your answer to four decimal places Consider the that fewer than 50 of licensed drivers prefer to keep complete control of the car when driving convincing if the probability that the event occurs when Ho p 0 5 is true is less than 0 05 the signficance level of 0 05 there is 0 convincing evidence that

The driving time for an individual from his home to his work is uniformly distributed between 200 to 470 seconds a Determine the probability density function f x 0 b Compute the probability that the driving time will be less than or equal to 415 seconds Round your answer to two decimal places c Determine the expected driving time in seconds sec 200 x 470 elsewhere d Compute the variance e Compute the standard deviation in seconds Round your answer to two decimal places sec MY NOTES

To indirectly measure the distance across a river Makayla stands on one side of the river and uses sight lines to a landmark on the opposite bank Makayla draws the diagram below to show the lengths and angles that she measured Find PR the distance across the river Round your answer to the nearest foot R 165 ft 235 ft E

length of stay in the ICU is 3 3 days Assume that this length of stay in the ICU has an exponential distribution Round your answers to four decimal places a What is the probability that the length of stay in the ICU is one day or less b What is the probability that the length of stay in the ICU is between two and three days c What is the probability that the length of stay in the ICU is more than five days Need Help Submit Answer Read It DETAILS ASWSBE14 6 TB 4 011 5 2 5 Points The time it takes a mechanic to change the oil in a car is exponentially distributed with a mean of 13 minutes MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

Which of the following are continuous random variables I the weight of an elephant II the time to answer a questionnaire III the number of floors in a skyscraper IV the square feet of countertop in a kitchen I and II only III and IV only O I II and IV only O I II III and IV X

Suppose the following estimated regression equation was determined to predict salary based on years of experience Estimated Salary 23 237 70 2277 01 Years of Experience What is the estimated salary for an employee with 15 years of experience

The following table gives the number of parking tickets obtained in a semester and the GPAS of 5 randomly selected drivers Number of Tickets 1 3 4 5 8 GPA 5 4 5 4 1 5 1 Calculate the coefficient of determination r2 Round your answer to three decimal places

1 In the game of roulette a player can place a 6 bet on the number 1 and have a 38 probability of winning If the metal ball lands on 1 the player gets to keep the 6 paid to play the game and the player is awarded an additional 210 Otherwise the player is awarded nothing and th casino takes the player s 6 What is the expected value of the game to the player If you played the game 1000 times how much would you expect to lose Note that the expected value is the amount on average one would expect to gain or lose each game The expected value is Round to the nearest cent as needed The player would expect to lose about Round to the nearest cent as needed

An investigator analyzed the leading digits from 792 checks issued by seven suspect companies The frequencies wer found to be 255 143 110 87 49 46 46 41 and 15 and those digits correspond to the leading digits of 1 2 3 4 5 6 7 8 and 9 respectively If the observed frequencies are substantially different from the frequencies expected with Benford s law shown below the check amounts appear to result from fraud Use a 0 01 significance level to test for goodness of fit with Benford s law Does it appear that the checks are the result of fraud Leading Digit Actual Frequency Benford s Law Distribution of Leading Digits Determine the null and alternative hypotheses Ho 1 255 30 1 H 2 3 4 5 6 143 110 87 49 46 17 6 12 5 9 7 7 9 6 7 At most three leading digits have frequencies that do not conform to Benford s law At least one leading digit has a frequency that does not conform to Benford s law At least two leading digits have frequencies that do not conform to Benford s law The leading digits are from a population that conforms to Benford s law 7 46 5 8 8 41 5 1 9 15 4 6 L

Fill in the blank In a In a we test the claim that different populations have the same proportions of some characteristics two way analysis of variance test of homogeneity we test the claim that different populations have the same proportions of some characteristi McNemar s test

The table below lists leading digits of 317 inter arrival Internet traffic times for a computer along with the frequencies of leading digits expected with Benford law When using these data to test for goodness of fit with the distribution described by Benford s law identify the null and alternative hypotheses Leading Digit Benford s Law Leading Digits of Inter Arrival Traffic Times 2 3 4 5 6 7 8 9 1 30 1 17 6 12 5 9 7 7 9 6 7 5 8 5 1 4 6 76 62 Choose the correct answer below 29 33 19 27 28 OA Ho None of the proportions are equal to the given claimed value H P 0 301 and p2 0 176 and p3 0 125 and and pg 0 046 OB Ho P 0 301 and p2 0 176 and p3 0 125 and and pg 0 046 H At least one of the proportions is different from the others 21 OC Ho At least one of the proportions is not equal to the given claimed value H P 0 301 and p 0 176 and p3 0 125 and and pg 0 046 O D Ho P 0 301 and p 0 176 and p3 0 125 and and pg 0 046 H At least one of the proportions is not equal to the given claimed value OE Ho At least one of the proportions is different from the others He 0301 and p 0176 and n 0 125 22

A is a table in which frequencies correspond to two variables is a table in which frequencies correspond to two variable Pearson correlation coefficient table contingency table standard normal distribution table

The 1 in 6 wins game As a special promotion for its 20 ounce bottles of soda a soft drink company printed a message on the inside of each bottle cap Some caps said Please try again while others said You re a winner The company advertised the promotion with the slogan 1 in 6 wins a prize Seven friends each buy one 20 ounce bottle at a local convenience store The store clerk is surprised when three of them win a prize Is this group of friends just lucky 3 out of 7 winners could happen by chance or is the company s 1 in 6 claim inaccurate 3 out of 7 winners is not likely to happen by chance First perform a simulation to help answer this question For now let s assume that the company is telling the truth and that every 20 ounce bottle of soda it fills has a 1 in 6 chance of getting a cap that says You re a winner 1 How can we model this 2 Define and conduct one trial for this simulation

The probability distribution of x the number of defective tires on a randomly selected automobile checked at a certain inspection station is given in the following table X p x 0 1 2 3 a Calculate the mean value of x Hx tires per car 4 0 55 0 18 0 05 0 05 0 17 b Interpret the mean value of x in the context of a long sequence of observations of number of defective tires tires per car In repeated inspections of cars at this station the mean number of defective tires is c What is the probability that x exceeds its mean value P x H d Calculate the standard deviation of x Round your answer to four decimal places tires per car Jy

Submit Answer 2 4 Points The size of the left upper chamber of the heart is one measure of cardiovascular health When the upper left chamber is enlarged the risk of heart problems is increased A paper described a stu which the left atrial size was measured for a large number of children ages 5 to 15 years Based on this data the authors concluded that for healthy children left atrial diameter was approximat normally distributed with a mean of 26 1 mm and a standard deviation of 4 1 mm Round your answers to four decimal places DETAILS PECKSTAT3 6 R 109 S USE SALT a Approximately what proportion of healthy children have left atrial diameters less than 24 mm b Approximately what proportion of healthy children have left atrial diameters greater than 32 mm c Approximately what proportion of healthy children have left atrial diameters between 25 and 30 mm d For healthy children what is the value in mm for which only about 20 have a larger left atrial diameter mm MY NOT

The Average American Is in Credit Card Debt No Matter the Economy reported that only 35 of credit card users pay off their bill every month Suppose that the reported perce was based on a random sample of 2 000 credit card users You are interested in learning about the value of p the proportion of all credit card users who pay off their bill every month The following table is similar to the table that appears in Examples 8 4 and 8 5 and is meant to summarize what you know about the sampling distribution of p in the situation just described What You Know information has been provided Complete the table by filling in the How You Know It column What You Know The sampling distribution of p is centered at the actual but unknown value of the population proportion An estimate of the standard deviation of p which describes how much the p values spread out around the population proportion p is 0 011 p p p P 0x P H p 0 5 H p X 0 35 1 0 35 2 000 0 35 1 0 35 2 000 0 35 1 0 35 2 000 0 35 1 0 35 2 000 10 3561035 How You Know It MY NOTES PRACTICE ANC

4 Suppose X is a continuous random variable with probability density function f x Express the shaded region as a probability statement 2 0 1 OP 0 X 1 OP 0 50 X 1 O P X 1 OP 0 x 0 50 Previous Answer d X

A newspaper story claims that more houses are purchased by singles now than singles 5 years ago To test this claim two studies were conducted on the buying habits of singles over the past 5 years In the first study 200 house purchases in the current year were randomly selected and 80 of those were made by singles In the second study again 200 house purchases were randomly selected from 5 years ago and 65 of those were made by single people Test the newspaper s claim using a 0 01 level of significance Is there sufficient evidence to support the newspaper s claim Let singles now be Population 1 and let singles 5 years ago be Population 2 Step 1 of 3 State the null and alternative hypotheses for the test Fill in the blank below Ho P1 P2 Ha P1

Isle Royale an Island in Lake Superior has provided an important study site of wolves and their prey Of special interest is the study of the number of moose killed by wolves In the period from 1958 to 1974 there were 296 moose deaths identified as wolf kills The age distribution of the kills is as follows Age of Moose in Years Calf 0 5 yr 1 5 6 10 11 15 16 20 USE SALT Number Killed by Wolves 105 50 77 58 6 a For each age group compute the probability that a moose in that age group is killed by a wolf Round your answers to four decimal places 0 5 1 5 6 10 11 15 16 20 b Consider all ages in a class equal to the class midpoint Find the expected age of a moose killed by a wolf and the standard deviation of the ages Round your answers to two decimal places U

Does it pay to ask for a raise A national survey of heads of households showed the percentage of those who asked for a raise and the percentage who got one According to the survey of the women interviewed 28 asked for a raise and of those women who had asked for a raise 45 received the raise If a woman is selected at random from the survey population of women find the following probabilities Enter your answers decimal places a P woman asked for a raise b P woman received raise given she asked for one c P woman asked for raise and received raise

Does it pay to ask for a raise A national survey of heads of households showed the percentage of those who asked for a raise and the percentage who got one According to the survey of the men interviewed 20 hi asked for a raise and 62 of the men who had asked for a raise received the raise If a man is selected at random from the survey population of men find the following probabilities Enter your answers to three decir places a P man asked for a raise b P man received raise given he asked for one c P man asked for raise and received raise

The Orchard Cafe has found that about 6 of the diners who make reservations don t show up If 78 reservations have been made how many diners can be expected to show up Find the standard deviation of this distribution Round your answers to two decimal places H 45 o 2 10

In an ESP experiment subjects must predict whether a number randomly generated by a computer will be odd or even Round your answer to four decimal pla a What is the probability that a subject would guess exactly 18 correct in a series of 36 trials b What is the probability that a subject would guess more than 20 correct in a series of 36 trials

The Orchard Cafe has found that about 6 of the diners who make reservations don t show up If 78 reservations have been made how many diners can be expected to show up Find the standard deviation of this distribution Round your answers to two decimal places 4 68 a 2 10

Consumer Reports rated airlines and found that 81 of the flights involved in the study arrived on time i e within 15 minutes of scheduled arrival time Assuming that the on time arrival rate is representative of the entire commercial airline industry consider a random sample of 206 flights Round your answers to two decimal places What is the expected number that will arrive on time 166 86 What is the standard deviation of this distribution 5 59

Internet tax In 2013 the Gallup Poll asked 1033 U S adults whether they believed that people should pay sales tax on items purchased over the internet Of these 434 said they supported such a tax Does the survey provide convincing evidence that less than 46 of U S adults favor an internet sales tax Use the a 0 10 level of significance and the P value method with the TI 84 Plus calculator Part 1 of 4 a State the null and alternate hypotheses HoP 46 46 This hypothesis test is a left tailed Alternate Answer Ho p 0 46 H p 0 46 This hypothesis test is a left tailed test Part 1 4 Part 2 of 4 test b Compute the P value Round the answer to at least four decimal places

The formula used to calculate a large sample confidence interval for p is P 1 P pa z critical value n What is the appropriate z critical value for each of the following confidence levels Round your answers to two decimal places a 95 b 96 c 83 Submit Answer LAUSE SALT

6 The disease status and exposure of a small population is summarized in the given table below An individual is randomly selected from this population Define A to be the event that the selected individual has been exposed and define event B to be the event that the selected individual has the disease Find P A B Exposure No Exposure Answer Disease 56 24 Previous Answer 0 7 No Disease 384 668

7 The disease status and exposure of a small population is summarized in the given table below An individual is randomly selected from this population Define A to be the event that the selected individual has been exposed and define event B to be the event that the selected individual has the disease Find the probability that the individual has the disease given they have not been exposed i e find P BIA Exposure No Exposure Disease Answer 64 18 No Disease Previous Answer none 404 Round your answer to the nearest thousandths 712

5 Assume each outcome is equally likely in the sample space S provided below Find P AB S 1 2 3 4 5 6 7 8 A 2 4 5 6 B 1 2 3 5 Do not round your answer Answer

Question 7 of 7 7 Suppose P A 25 P B 0 30 and P AB 0 15 Find P AUB Answer

Question 5 3 Assuming each outcome is equally likely in the sample space find P AUB FOB Answer

A company estimates that 0 8 of their products will fail after the original warranty period but within 2 years of the purchase with a replacement cost of 100 If they offer a 2 year extended warranty for 8 what is the company s expected value of each warranty sold

Find the expected value for the random variable X 3 1 3 9 12 P X x 0 22 0 14 0 23 0 2 0 21 X O 3 8 2 69 O4 35 01 97 O 4 49 O 5 15

Suppose you decided to play a gambling game order to play the game there is a 1 50 dollar fee to play If you roll a 1 2 or 3 you win nothing i e your net profit is 1 5 dollars If you roll a 4 or 5 you win 4 25 i e your net profit is 2 75 If you roll a 6 you win 4 50 i e your net profit is 3 a Use the information described above to constuct a probability distribution table for the random variable which represents the net profit of your winnings Note Be sure to enter your probabilities as reduced fractions X You roll a 1 2 or 3 You roll a 4 or 5 You roll a 6 P x You roll a 1 2 or 3 You roll a 4 or 5 You roll a 6 b Find the amount you would expect to win or lose each time you played the game Round your final answer to two decimal places

For families with 5 children let X be the number of children with Genetic Condition B Can the following table be a probability distribution for the random variable X x P x 1 0 2332 2 0 0311 3 0 114 4 0 5199 0 1018 LO 5 O no Q yes