Name _______________________________
Period _______________
Special Topics
Section 8.5 Homework
1. Linda is a sales associate at a large auto dealership. She expects to earn $400 for each vehicle she
sells. Linda motivates herself by using probability estimates of her sales. For a sunny Saturday in
April, she estimates her sales as follows:
Give the probability model for Linda's earnings. What are her mean earnings?
2. Below is a probability model for the grade of a randomly chosen student in Statistics 101 at North
Carolina State University, using the 4-point scale. What is the mean grade in this course? What is the
standard deviation of the grades?
3. Below gives probability models for the number of rooms in owner-occupied and rented housing
units. Find the mean number of rooms for each type of housing. Make probability histograms for the
two models and mark the mean on each histogram. You see that the means describe an important
difference between the two models: Owner-occupied units tend to have more rooms.
4. Typographical and spelling errors can be either “non-word errors” or “word errors.” A non-word
error is not a real word, as when “the” is typed as “teh.” A word error is a real word, but not the right
word, as when “lose” is typed as “loose.” When undergraduates are asked to write a 250-word essay
(without spell-checking), the number of non-word errors has this probability model:
The number of word errors has this model:
What are the mean numbers of non-word errors and word errors in an essay? How does the
difference between the means describe the difference between the two models?
5. The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire
destroying your home. Insurance spreads the risk: We all pay a small amount, and the insurance policy
pays a large amount to those few of us whose homes burn down. An insurance company looks at the
records for millions of homeowners and sees that the mean loss from fire in a year is μ = $250 per
person. (The great majority of us have no loss, but a few lose their homes. The $250 is the average
loss.) The company plans to sell fire insurance for $250 plus enough to cover its costs and profit.
Explain clearly why it would be unwise to sell only 12 policies. Then explain why selling thousands of
such policies is a safe business.
6. Should you buy the extended warranty on a new washing machine? Suppose there are two outcomes
— an 85% probability of needing no repairs, and a 15% probability of needing a $200 repair during the
warranty period. Based on the mean outcome for this model, what would be a “break-even” price to
you for the extended warranty? (The company, of course, will charge more than this in order to make
a profit.)
7. An American roulette wheel has 38 slots numbered 0, 00, and 1 to 36. The ball is equally likely to
come to rest in any of these slots when the wheel is spun. The slot numbers are laid out on a board on
which gamblers place their bets. One column of numbers on the board contains multiples of 3—that is,
3, 6, 9, …, 36. Joe places a $1 “column bet” that pays out $3 if any of these numbers comes up.
1.
What is the probability model for the outcome of one bet, taking into account the $1 cost of
a bet?
2. What are the mean and standard deviation for this model?
3. Joe plays roulette every day for years. What does the law of large numbers tell us about his
results?
8. This table shows the prizes and respective probabilities for a lottery:
On average, how much money from a $1 ticket comes back to you in prizes?
9. A state lottery “Pick 3” game offers a choice of several bets. You choose a three-digit number and
bet $1. The lottery commission announces the winning three-digit number, chosen at random, at the
end of each day. The “box” pays $82.33 if the number you chose has the same digits as the winning
number, in any order. Otherwise, you lose your dollar. Find the mean winnings for a bet on the box,
taking into account that you paid $1 to play. (Assume that you chose a number having three distinct
digits.)
10. Suppose a test is designed in which each question has 5 possible answer choices (ABCDE).
1.
If you get + 1 point for every correct answer, what would the “penalty” for a wrong answer
need to be if you wanted guessing to neither help nor hurt the score on average? (Hint:
Consider a test with 5 questions on it.)
2. If you are able to eliminate some but not all of the wrong answers, does it help on average to
guess among the remaining choices