ProblemUID,ProblemTitle,Description,distribution,parameter(s),hint,comment
1,Broken Yardstick P1,A yard stick is broken randomly into 8 pieces. X = the distance (in yards) from one end of the stick to the 3rd break point.,Beta,alpha = 3; beta = 6,break points are uniform over the stick,
2,Software Training P1,Four of the 15 administrative assistants in an office have been trained in a new software program. Five of the 15 assistants are randomly selected to take part in a focus group about office procedures. X = the number of those in the focus group who have had the software training.,Hypergeometric,N=15; M= 4; n=5,focus group sampled without replacement,
3,Toothpaste Use P1,A 6.25-ounce tube of Crest toothpaste is used regularly by all members of a household until the tube is finished and then it is replaced by another tube of the same brand. X = how much toothpaste (in ounces) is in the current tube at a randomly chosen time.,Continuous Uniform,A=0; B=6.25,toothpaste used up at a constant pace,
4,Correct Change P1,X = how many of the next 20 customers at a hot dog stand have the correct change to pay for their purchases.,Binomial,n = 20; p = probability that a random customer has the correct change,customers are essentially independent,Also approximately normal if p is not too small or too large
5,Lemon Law P1,"Ohio's lemon law allows people who buy a lemon (a vehicle that turns out to be excessively defective) to return it for a refund. However, in Ohio, only about 24 cars are expected to be declared ""lemons"" each year. X = the time until the next car is returned under Ohio's lemon law.",Exponential,r = 24 per year (mean = 1/r=1/24 year or 1/2 month),time to rare event,alternate answer is Gamma with parameters 1 and 1/24
6,Powerball Lottery P1,X = the first number drawn (from the numbers 1 to 59) for this week's Powerball lottery.,Discrete Uniform,a=1; b=59,equal chances on numbers,
7,Survey Responder P1,"A polling company gets respondents for a survey through random digit dialing. However there is a large proportion of people, who are not available or refuse to answer the survey questions. X=how many calls must be made before they get the first respondent. ",Geometric,p = probability that a person is available and willing to be interviewed when called,calls until first success,
8,Survey Responder P2,"A polling company gets respondents for a survey through random digit dialing. However there is a large proportion of people, who are not available or refuse to answer the survey questions. The company needs a sample of 100 respondents in order to fulfill a contract. X=how many calls are made before they fulfill their contract.",Negative Binomial,k = 100; p = probability that a person is available and willing to be interviewed when called,calls until 100th success,Also approximately normal if p is not too small
9,GW Bridge Traffic P1,"The George Washington Bridge connecting New Jersey and New York is the busiest bridge in the world, carrying about 100 million vehicles a year. On average, there is approximately one fatal accident per 20 million vehicles. X = the number of fatal accidents on the George Washington Bridge next year.",Poisson,lamda = 5,fatal accidents are rare and close to independent of each other,Poisson still reasonable even if the number of fatal accidents changes with the seasons of the year.
10,B-negative Blood Type P1,X = How many people must be tested until you find 3 that have a B negative blood type. ,Negative Binomial,k = 3; p = probability that a random person has B- blood (about 2%),tests until 3rd success,
11,Die Rolls P1,X = how many of each number you get when you roll a die 25 times,Multinomial,n = 25; p1=p2=p3=p4=p5=p6 = 1/6,counting multiple categories,
12,DNA Radiation Damage P1,A radiation source causes the strand of DNA along a chromosome to randomly break into two pieces (i.e. one break point). X = the proportion of the way from one end of the strand to the other that the break occurs.,Continuous Uniform,A=0; B=1,probability spread evenly,a.k.a. Standard Uniform
13,Miami Homicide P2,The homicide rate in Miami Florida averages 1 per week throughout the year. X = the day when Miami will see it's 10th homicide next year.,Gamma,k = 10; theta = 7,time to 10th rare event, a.k.a. Erlang distribution in this case
14,Extra Large Eggs P1,"The USDA defines an ""Extra Large"" egg as one weighing between 2.25 and 2.5 ounces. Eggs are packaged by the dozen with 30 dozen to a case. X = the total weight (ounces) of the eggs in a random case delivered to a grocery.",Normal,mu = 855 ounces; sigma = 1.37 ounces,sum of large number of independent comparable parts,Assume relatively uniform distribution of weights between 2.25 and 2.5 ounces per egg . Otherwise can get bounds on mu and sigma that will be close to these values.
15,B-negative Blood Type P2,X = How many people must be tested until you find someone with a B negative blood type. ,Geometric,p = probability that a random person has B- blood (about 2%),tests until 1st success,
16,GW Bridge Traffic P2,"The George Washington Bridge connecting New Jersey and New York is the busiest bridge in the world, carrying about 100 million vehicles a year. On average, there is approximately one fatal accident per 20 million vehicles. X = the time (in years) until the next fatal accident on the George Washington Bridge.",Exponential,r = 5 per year (mean = 1/r = 1/5 year),time to rare event,Assumption of constant rate may not be appropriate (affects the mean - but exponential distribution still likely to be reasonable)
17,Penn Turnpike Call-boxes P1,Emergency call-boxes are systematically placed 1 mile apart along the Pennsylvania Turnpike where cell phone service is very spotty. Your car breaks down suddenly as you are driving east on the Pennsylvania Turnpike though luckily you had been keeping an eye out for the call boxes before your breakdown. X= the distance you will need to walk to reach the closest call-box.,Continuous Uniform,A = 0; B = 0.5,no preference for any particular place to break down,
18,Penn Turnpike Call-boxes P2,Emergency call-boxes are systematically placed 1 mile apart along the Pennsylvania Turnpike where cell phone service is very spotty. Your car breaks down as you are driving east on the Pennsylvania Turnpike so you walk eastward until you reach a call-box. X= the distance in miles you will need to walk to reach the next call-box.,Continuous Uniform,A=0; B=1,no preference for any particular place to break down,"alternate answer is Beta (1,1) a.k.a. Standard Uniform"
19,B-negative Blood Type P3,X = How many people have a B negative blood type in a sample of just one randomly selected person. ,Bernoulli,p = probability that a random person has B- blood (about 2%),variable can only be zero or one,alternate answer is Binomial with n=1
20,2012 Presidential Race P1,There were 29 national polls published in the week before the 2012 U.S. Presidential race to gauge the voter support for President Barrack Obama and former Massachusetts Governor Mitt Romney. Before the results of these polls were known: let X = the average percentage of the two-party vote amongst respondents that favor the President's bid for re-election across the 29 polls.,Normal,mu = percentage of the two-party vote who favored the President's re-election for all voters; sigma^2 = sum of variances for individual polls/841,each of the 29 individual polls had a large sample size,assumes the average bias across all polls is negligble
21,Broken Wikipedia Links P1,X = the number of broken links in a randomly selected page in Wikipedia,Poisson,lamda = average number of broken links per page in all of Wikipedia,broken links are rare,
22,HS Pizza Delivery P1,20 high school students apply for a summer job delivering pizza and 12 of them own a car. Five of the students are picked randomly and called in for an interview. X = how many of the interviewed students own a car.,Hypergeometric,N=20; M=12; n=5,interviewees sampled without replacement,
23,Account Auditing Case P1,A small auditing firm does random checks on accounts and assigns cases that show irregularities to an accountant for further study. X = how many accounts are checked before all five of the firm's accountants have a case for review.,Negative Binomial,k = 5; p = probability that a random account shows irregularities,checks until 5th success,
24,CPU Job Processing P1,"Ten % of the jobs run on a computer take up 90% of the processing time and 10% of those top 10% take up 90% of the time dedicated to those jobs, etc. X= the time taken by a randomly selected job.",Pareto,k = log(10)/log(9),compare with 80-20 rule,
25,Petrol Pump Purchases P1,X = the number of gallons of gas purchased by the next 100 customers at a service station,Normal,mu = 100*average amount purchased; sigma=10*st.dev of amounts purchased,sum of gas purchased by independent customers,
26,Cook premature infant weight P1,X= the number of premature infants weighing less than 4 pounds are born tomorrow at the Cook County Hospital maternity ward in Chicago,Poisson,mu = average number of premature infants per day,number of rare events in a fixed time period,
27,Online hardware sales P1,"An online store only sells desktop computers, laptops, and tablets. X = how many of each type of machine will be in their next 100 sales",Multinomial,n=100; p1=probability that the next machine sold is a desktop; p2=probability it's a laptop; p3=1-p1-p2,counting multiple categories,
28,Customer Service Calls P1,X = the difference between the time to the next call to a customer service line and the time to the call after that.,Laplace,mu = 0; b = average time between calls,difference between X1= time to event governed by large independently acting population and independent X2 = time to same type of event,
29,Plane Crash intervals P1,X = the ratio of the time to the next major plane crash and the time between now and the one after that.,Continuous Uniform,A=0; B=1,time between plane crashes is nearly memoryless,a.k.a. Standard Uniform
30,Highway Car Colors P1,X = how many of the next 20 cars that pass you on the highway are silver colored,Binomial,n=20; p= population proportion of passing cars that are silver,car colors are independent,
31,Highway Car Colors P2,X = how many cars pass you on the highway before one of them is silver colored,Geometric,p= population proportion of passing cars that are silver,each passing car is another trial,
32,Restaurant salt use P1,A popular restaurant is completely full every Friday night. X = the ratio of the variance of the amount of salt the restaurant will use on Friday nights this month compared to next month.,F,df1 = number of Fridays this month minus 1; df2 = number of Fridays next month minus 1,What distribution models total salt use from Friday to Friday?,
33,Restaurant salt use P2,A popular restaurant is completely full every Friday night. X = the amount of salt the restaurant will use next Friday night.,Normal,mu = average amount used on Friday nights; sigma = standard deviation of amounts from Friday-to-Friday,sum of amounts used by approximately constant number of nearly independent customers,
34,Customer Service Calls P2,X = the number of calls to a customer service line in the next hour.,Poisson,mu = call rate per hour,number of events governed by large independently acting population,
35,Horse Race P1,Three horses are randomly picked from the 12 competing in a race and you place a bet on each of these three horses to win the race. X=the number of winning tickets purchased.,Bernoulli,p = 25% (3/12),only one winner possible,
36,Horse Race P2,"Three horses are randomly picked from the 12 competing in a race and you place a bet on each of these three horses to show (i.e. come in first, second, or third). X=the number of winning tickets purchased.",Hypergeometric,N=12; M=3; n=3,chosen horses each hold separate position,randomly picked means srs here
37,Alarm Clock P1,When you wake up tomorrow morning you look at the digital clock next to your bed and watch it until the time changes. X = how long you have to wait in minutes,Continuous Uniform,A=0; B=1,no time within the minute you awaken is more likely than another,a.k.a. Standard Uniform
38,Happiness Experiment P1,"40 subjects taking part in an experiment on the effect of exercise on happiness record how many times they laugh during the day before and the day after taking part in an active exercise program. Assuming the program has no effect on happiness, let X = the average difference between the recorded laughter values before and after the program.",Normal,mu =0; sigma^2 = variance of difference for individuals divided by 40,with no effect expect no difference,assume of independent individuals
39,Happiness Experiment P2,"80 subjects taking part in an experiment on the effect of exercise on happiness record how many times they laugh during the day before and the day after either taking part in an education program on the value of exercise (40 random subjects) or taking part in an active exercise program (40 random subjects). Assuming neither program has an effect on happiness, let X = the ratio of the average difference between the recorded before and after laughter values for those in the education program versus the average difference for those in the exercise program. ",Cauchy,parameters 0 and 1,CLT for numerator and denominator,note same variance for numerator and denominator (a.k.a. standard Cauchy)
40,Miami Homicide P1,The homicide rate in Miami Florida averages 1 per week throughout the year. X = the day when Miami will see it's 1st homicide next year.,Exponential,r = 1/7 per day (mean = 7 days or 1 week),time to first rare event,
41,Water Usage P1,X = the amount of water used next September by a person taking a daily shower,Normal,mu = average per day times 30; sigma = standard deviation of total,Sum of usage over 30 days in month,
42,Penn Turnpike Call-boxes P3,Emergency call-boxes are systematically placed 1 mile apart along the Pennsylvania Turnpike where cell phone service is very spotty. Five cars break down driving east on the Pennsylvania Turnpike and each driver walks eastward until they reach a call-box. X= the median distance in miles these five need to walk to reach the next call-box.,Beta,alpha = 3; beta = 3,third largest of five with no preference for any particular place to break down,
43,Transmission failure P1,"For automatic transmissions in city buses that are still in service, the rate at which they fail increases linearly as they wear out with age. X = how long a transmission will last.",Weibull,k = 2; lamda = modal time to failure,failure times will linear hazard rate,
44,Petrol Pump Purchases P2,X = how many of the next 10 customers at a service station buy their gas with a credit card,Binomial,n = 10; p = proportion of all customers using a credit card,dichotomous population of independent customers,
45,Lemon Law P2,"Ohio's lemon law allows people who buy a lemon (a vehicle that turns out to be excessively defective) to return it for a refund. However, in Ohio, only about 24 cars are expected to be declared ""lemons"" each year. X = how many cars are returned in a random month under Ohio's lemon law.",Poisson,lamda = 2,lemons are rare and nearly independent,
46,Color Blind P1,"When ten men must be studied before two are found to be color blind, let X = how many were studied before the first color blind man was found",Discrete Uniform,a=1; b=9,no preference for any particular trial,
47,GW Bridge Traffic P3,"The George Washington Bridge connecting New Jersey and New York is the busiest bridge in the world, carrying about 100 million vehicles a year. On average, there is approximately one fatal accident per 20 million vehicles. X = the time (in years) until there are 5 more fatal accidents on the George Washington Bridge.",Gamma,k =5 ; theta = 5,time to 5th rare event,Assumption of constant rate may not be appropriate (affects the mean - but Gamma distribution still likely to be reasonable)
48,Cafeteria Utensils P1,"A cafeteria provides teaspoons, soupspoons, non serrated and serrated knives, and forks for customers to take as needed. X = how many soupspoons are taken by the next 50 customers.",Binomial,n=50; p= probability a customer uses a soupspoon,customers are independent,Also assume that customers don't take more than one (students should be able to recognize that if the problem asked about X = how many of each kind are taken by the next 50 customers it would NOT be multinomial since individual customers take more than one utensil)
49,Jury Pool P1,When a judge of the Franklin County Court of Common Pleas issues a call to the courtroom to the Cleark of Courts - the clerk will ranomly select a jury panel of 24 people for the trial. Suppose the current jury pool consists of 30 women and 20 men. X = the number of women in the selected jury panel.,Hypergeometric,N=50; M=30; n=24,Count in without replacement sample,
50,DNA Radiation Damage P1,A radiation source causes the strand of DNA along a chromosome to randomly break into seven pieces (i.e. six break points). X = the proportion of the way from one end of the strand to the other that the second break occurs.,Beta,alpha = 2; beta = 6,break points uniform over strand,
51,Salmon location P1,An increasing temperature gradient in a river radiates from the coldest/deepest part of the channel and the Sockeye salmon swimming in the river roughly follow the colder parts of this gradient as they move upstream. X= proportion of the way from one bank to the other that a random salmon can be found.,Beta,alpha and beta > 1,unimodal model for proportions,other unimodal models may be more appropriate depending on nature of riverbed
52,Ice Usage P1,"X = how much ice will be used on the next full Southwest Airline flight from Denver, CO to Raleigh/Durham, NC",Normal,mu = average ice used on full flights; sigma = flight to flight st. dev.,total used for fixed number of nearly independent passengers,
53,Prius Traffic P1,X = how long until the next 2001 Toyota Prius hybrid goes through a toll booth on the Golden Gate Bridge,Exponential,r = rate that such cars are seen,memoryless,
54,Left handers P1,X = how many randomly chosen people will it take to find 10 who are left-handed,Negative Binomial,k = 10; p about 0.11,people until 10th success,
55,Left handers P2,X = how many randomly chosen people will it take find one who is left-handed,Geometric,p about 0.11,,
56,Plant breeding P1,X = the number of red-flowering plants in 100 crosses between pink flowering plants,Binomial,n = 100; p = 0.25 ,count in 100 trials,assume Mendelian genetics
57,Fish tank P1,X = the position (degrees clockwise from you) of a fish swimming in a cylindrical tank.,Continuous Uniform,A = 0; B = 360,no preferece for any angle,
58,Fish tank P2,X = median position of the same fish wrt the first person visiting the aquarium each day this week.,Beta,alpha = 4; beta = 4,4th largest of 7,
59,NYT Crossword P1,X = the average amount of time that a sample of 500 regular subscribers to the New York Times spend on the Sunday crossword puzzle,Normal,mu = average for all NYT subscribers; sigma = person-to-person standard deviation,average of 500 independent tries,population distribution would be very skewed
60,Prius Traffic P2,X = how many 2001 Toyota Prius hybrids go across the Golden Gate bridge tomorrow,Poisson,lamda = average number per day,number of independent rare events,
61,Internet purchases P1,X = In a sample of just one person; how many favor imposing a uniform sales tax on internet purchases.,Bernoulli,p = proportion in population that favor the proposal,value can be 1 or 0,
62,Vending Machine P1,X = how much time until the next diet coke is purchased from a vending machine.,Exponential,r = rate that diet coke purchasers come to the machine,customer arrivals basically memoryless,
63,Vending Machine P2,X = how much time until five separate people purchase a diet coke from a vending machine,Gamma,k =5; theta = time between separate purchases,Waiting time until 5th event,
64,Tips P1,X = how many of the next ten parties that a waiter serves will leave a tip of more than 20% of the bill. ,Binomial,n = 10; p = proportion of all parties leaving that sized tip,number of indpendent parties out of 10,
65,Birth Month P1,X = the birth month of the next person to board a plane from Honolulu to Los Angeles,Discrete Uniform,a = 1; b = 12,all months equally likely,