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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 \(\mathrm{A}\) 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?

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?

A manufacturer ships toasters in cartons of 20 . In each carton, they estimate a \(5 \%\) chance that one of the toasters will need to be sent back for minor repairs. What is the probability that in a carton, there will be exactly 3 toasters that need repair?

A soccer team estimates that they will score on \(8 \%\) of the corner kicks. In next week's game, the team hopes to kick 15 corner kicks. What are the chances that they will score on 2 of those opportunities?

A cable provider wants to contact customers in a particular telephone exchange to see how satisfied they are with the new digital TV service the company has provided. All numbers are in the 452 exchange, so there are 10,000 possible numbers from \(452-0000\) to \(452-9999 .\) If they select the numbers with equal probability: a. What distribution would they use to model the selection? b. The new business "incubator" was assigned the 200 numbers between \(452-2500\) and \(452-2699,\) but these businesses don't subscribe to digital TV. What is the probability that the randomly selected number will be for an incubator business? c. Numbers above 9000 were only released for domestic use last year, so they went to newly constructed residences. What is the probability that a randomly selected number will be one of these?

Lifetimes of electronic components can often be modeled by an Exponential model. Suppose quality control engineers want to model the lifetime of a hard drive to have a mean lifetime of 3 years. a. What value of \(\lambda\) should they use? b. With this model, what would the probability be that a hard drive lasts 5 years or less?

Suppose occurrences of sales on a small company's website are well modeled by a Poisson model with \(\lambda=5 /\) hour. a. If a sale just occurred, what is the expected waiting time until the next sale? b. What is the probability that the next sale will happen in the next 6 minutes?

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.

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.

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.