Problem 1
Do these situations involve Bernoulli trials? Explain. a) We roll 50 dice to find the distribution of the number of spots on the faces. b) How likely is it that in a group of 120 the majority may have Type A blood, given that Type A is found in \(43 \%\) of the population? c) We deal 7 cards from a deck and get all hearts. How likely is that? d) We wish to predict the outcome of a vote on the school budget, and poll 500 of the 3000 likely voters to see how many favor the proposed budget. e) A company realizes that about \(10 \%\) of its packages are not being sealed properly. In a case of 24, is it likely that more than 3 are unsealed?
Problem 2
Do these situations involve Bernoulli trials? Explain. a) You are rolling 5 dice and need to get at least two 6 's to win the game. b) We record the distribution of eye colors found in a group of 500 people. c) A manufacturer recalls a doll because about \(3 \%\) have buttons that are not properly attached. Customers return 37 of these dolls to the local toy store. Is the manufacturer likely to find any dangerous buttons? d) A city council of 11 Republicans and 8 Democrats picks a committee of 4 at random. What's the probability they choose all Democrats? e) A 2002 Rutgers University study found that \(74 \%\) of high school students have cheated on a test at least once. Your local high school principal conducts a survey in homerooms and gets responses that admit to cheating from 322 of the 481 students.
Problem 3
Think about the Tiger Woods picture search again. You are opening boxes of cereal one at a time looking for his picture, which is in \(20 \%\) of the boxes. You want to know how many boxes you might have to open in order to find Tiger. a) Describe how you would simulate the search for Tiger using random numbers. b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities that you might find your first picture of Tiger in the first box, the second, etc. d) Calculate the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.
Problem 4
You are one space short of winning a child's board game and must roll a 1 on a die to claim victory. You want to know how many rolls it might take. a) Describe how you would simulate rolling the die until you get a 1 . b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities that you might win on the first roll, the second, the third, etc. d) Calculate the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.
Problem 6
Suppose \(75 \%\) of all drivers always wear their seatbelts. Let's investigate how many of the drivers might be belted among five cars waiting at a traffic light. a) Describe how you would simulate the number of seatbelt-wearing drivers among the five cars. b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities there are no belted drivers, exactly one, two, etc. d) Find the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.
Problem 7
A Department of Transportation report about air travel found that, nationwide, \(76 \%\) of all flights are on time. Suppose you are at the airport and your flight is one of 50 scheduled to take off in the next two hours. Can you consider these departures to be Bernoulli trials? Explain.
Problem 8
A Department of Transportation report about air travel found that airlines misplace about 5 bags per 1000 passengers. Suppose you are traveling with a group of people who have checked 22 pieces of luggage on your flight. Can you consider the fate of these bags to be Bernoulli trials? Explain.
Problem 9
A basketball player has made \(80 \%\) of his foul shots during the season. Assuming the shots are independent, find the probability that in tonight's game he. a) misses for the first time on his fifth attempt. b) makes his first basket on his fourth shot. c) makes his first basket on one of his first 3 shots.
Problem 10
Suppose a computer chip manufacturer rejects \(2 \%\) of the chips produced because they fail presale testing. a) What's the probability that the fifth chip you test is the first bad one you find? b) What's the probability you find a bad one within the first 10 you examine?
Problem 13
Raaj works at the customer service call center of a major credit card bank. Cardholders call for a variety of reasons, but regardless of their reason for calling, if they hold a platinum card, Raaj is instructed to offer them a double-miles promotion. About \(10 \%\) of all cardholders hold platinum cards, and about \(50 \%\) of those will take the double-miles promotion. On average, how many calls will Raaj have to take before finding the first cardholder to take the doublemiles promotion?
