# RANDOM VARIABLES ON THE COMPUTER EXERCISES. 9. Variation 1. Find the standard deviations of the random

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1 Exercises 383 RANDOM VARIABLES ON THE COMPUTER Statistics packages deal with data, not with random variables. Nevertheless, the calculations needed to find means and standard deviations of random variables are little more than weighted means. Most packages can manage that, but then they are just being overblown calculators. For technological assistance with these calculations, we recommend you pull out your calculator. EXERCISES 1. Expected value. Find the expected value of each random variable: a) b) x P(X=x) x P(X = x) Expected value. Find the expected value of each random variable: a) x P(X = x) b) x P(X = x) Pick a card, any card. You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win \$5. For any club, you win \$10 plus an extra \$20 for the ace of clubs. a) Create a probability model for the amount you win. b) Find the expected amount you ll win. c) What would you be willing to pay to play this game? 4. You bet! You roll a die. If it comes up a 6, you win \$100. If not, you get to roll again. If you get a 6 the second time, you win \$50. If not, you lose. a) Create a probability model for the amount you win. b) Find the expected amount you ll win. c) What would you be willing to pay to play this game? 5. Kids. A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. (Assume boys and girls are equally likely.) a) Create a probability model for the number of children they might have. b) Find the expected number of children. c) Find the expected number of boys they ll have. 6. Carnival. A carnival game offers a \$100 cash prize for anyone who can break a balloon by throwing a dart at it. It costs \$5 to play, and you re willing to spend up to \$20 trying to win. You estimate that you have about a 10% chance of hitting the balloon on any throw. a) Create a probability model for this carnival game. b) Find the expected number of darts you ll throw. c) Find your expected winnings. 7. Software. A small software company bids on two contracts. It anticipates a profit of \$50,000 if it gets the larger contract and a profit of \$20,000 on the smaller contract. The company estimates there s a 30% chance it will get the larger contract and a 60% chance it will get the smaller contract. Assuming the contracts will be awarded independently, what s the expected profit? 8. Racehorse. A man buys a racehorse for \$20,000 and enters it in two races. He plans to sell the horse afterward, hoping to make a profit. If the horse wins both races, its value will jump to \$100,000. If it wins one of the races, it will be worth \$50,000. If it loses both races, it will be worth only \$10,000. The man believes there s a 20% chance that the horse will win the first race and a 30% chance it will win the second one. Assuming that the two races are independent events, find the man s expected profit. 9. Variation 1. Find the standard deviations of the random variables in Exercise Variation 2. Find the standard deviations of the random variables in Exercise Pick another card. Find the standard deviation of the amount you might win drawing a card in Exercise The die. Find the standard deviation of the amount you might win rolling a die in Exercise Kids again. Find the standard deviation of the number of children the couple in Exercise 5 may have. 14. Darts. Find the standard deviation of your winnings throwing darts in Exercise Repairs. The probability model below describes the number of repair calls that an appliance repair shop may receive during an hour. Repair Calls Probability a) How many calls should the shop expect per hour? b) What is the standard deviation?

2 384 CHAPTER 16 Random Variables 16. Red lights. A commuter must pass through five 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. X = # of red P(X = x) a) How many red lights should she expect to hit each day? 17. Defects. A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips, etc.). While none had more than three of these defects, 7% had three, 11% two, and 21% one defect. Find the expected number of appearance defects in a new car and the standard deviation. 18. Insurance. An insurance policy costs \$100 and will pay policyholders \$10,000 if they suffer a major injury (resulting in hospitalization) or \$3000 if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2000 policyholders may have a major injury, and 1 in 500 a minor injury only. a) Create a probability model for the profit on a policy. b) What s the company s expected profit on this policy? 19. Cancelled flights. Mary is deciding whether to book the cheaper flight home from college after her final exams, but she s unsure when her last exam will be. She thinks there is only a 20% chance that the exam will be scheduled after the last day she can get a seat on the cheaper flight. If it is and she has to cancel the flight, she will lose \$150. If she can take the cheaper flight, she will save \$100. a) If she books the cheaper flight, what can she expect to gain, on average? b) What is the standard deviation? 20. Day trading. An option to buy a stock is priced at \$200. If the stock closes above 30 on May 15, the option will be worth \$1000. If it closes below 20, the option will be worth nothing, and if it closes between 20 and 30 (inclusively), the option will be worth \$200. A trader thinks there is a 50% chance that the stock will close in the range, a 20% chance that it will close above 30, and a 30% chance that it will fall below 20 on May 15. a) Should she buy the stock option? b) How much does she expect to gain? c) What is the standard deviation of her gain? 21. Contest. You play two games against the same opponent. The probability you win the first game is 0.4. If you win the first game, the probability you also win the second is 0.2. If you lose the first game, the probability that you win the second is 0.3. a) Are the two games independent? Explain. b) What s the probability you lose both games? c) What s the probability you win both games? d) Let random variable X be the number of games you win. Find the probability model for X. e) What are the expected value and standard deviation? 22. Contracts. Your company bids for two contracts. You believe the probability you get contract #1 is 0.8. If you get contract #1, the probability you also get contract #2 will be 0.2, and if you do not get #1, the probability you get #2 will be 0.3. a) Are the two contracts independent? Explain. b) Find the probability you get both contracts. c) Find the probability you get no contract. d) Let X be the number of contracts you get. Find the probability model for X. e) Find the expected value and standard deviation. 23. Batteries. In a group of 10 batteries, 3 are dead. You choose 2 batteries at random. a) Create a probability model for the number of good batteries you get. b) What s the expected number of good ones you get? 24. Kittens. In a litter of seven kittens, three are female. You pick two kittens at random. a) Create a probability model for the number of male kittens you get. b) What s the expected number of males? 25. Random variables. Given independent random variables a) 3X b) Y + 6 c) X + Y d) X - Y e) X 1 + X Random variables. Given independent random variables a) X - 20 b) 0.5Y c) X + Y d) X - Y e) Y 1 + Y Random variables. Given independent random variables a) 0.8Y b) c) d) e) 2X X + 2Y 3X - Y Y 1 + Y Random variables. Given independent random variables a) 2Y + 20 b) 3X c) 0.25X + Y d) X - 5Y e) X 1 + X 2 + X 3 X 10 2 Y 20 5 X Y 12 3 X Y Mean 29. Eggs. A grocery supplier believes that in a dozen eggs, the mean number of broken ones is 0.6 with a standard SD X Y 12 3

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