Bluman A.G. Elementary Statistics- A Step By Step Approach

268 Chapter 5 Discrete Probability DistributionsNumber of jobs 1 2 3 4Probability 0.2 0.3 0.4 0.1Find the contractor’s expected profit per month. $720013. Rolling Dice If a person rolls doubles when she tossestwo dice, she wins $5. For the game to be fair, howmuch should she pay to play the game? $0.8314. Dice Game A person pays $2 to play a certain game byrolling a single die once. If a 1 or a 2 comes up, theperson wins nothing. If, however, the player rolls a 3, 4,5, or 6, he or she wins the difference between thenumber rolled and $2. Find the expectation for thisgame. Is the game fair? 33.3 cents; no15. Lottery Prizes A lottery offers one $1000 prize, one$500 prize, and five $100 prizes. One thousand ticketsare sold at $3 each. Find the expectation if a personbuys one ticket. $1.0016. In Exercise 15, find the expectation if a person buys twotickets. Assume that the player’s ticket is replaced aftereach draw and that the same ticket can win more thanone prize. $2.0017. Winning the Lottery For a daily lottery, a personselects a three-digit number. If the person plays for $1,she can win $500. Find the expectation. In the samedaily lottery, if a person boxes a number, she will win$80. Find the expectation if the number 123 is playedfor $1 and boxed. (When a number is “boxed,” it canwin when the digits occur in any order.) $0.50, $0.5218. Life Insurance A 35-year-old woman purchases a$100,000 term life insurance policy for an annualpayment of $360. Based on a period life table for theU.S. government, the probability that she will survivethe year is 0.999057. Find the expected value of thepolicy for the insurance company. $265.7019. Roulette A roulette wheel has 38 numbers, 1 through36, 0, and 00. One-half of the numbers from 1 through36 are red, and the other half are black; 0 and 00 aregreen. A ball is rolled, and it falls into one of the38 slots, giving a number and a color. The payoffs(winnings) for a $1 bet are as follows:?Red or black $1 0 $35Odd or even $1 00 $351–18 $1 Any single number $359–36 $1 0 or 00 $17If a person bets $1, find the expected value for each.a. Red 5.26 cents d. Any single number 5.26 centsb. Even 5.26 cents e. 0 or 00 5.26 centsc. 00 5.26 centsExtending the Concepts20. Rolling Dice Construct a probability distribution forthe sum shown on the faces when two dice are rolled.Find the mean, variance, and standard deviation of thedistribution. 7; 5.8; 2.421. Rolling a Die When one die is rolled, the expectedvalue of the number of spots is 3.5. In Exercise 20, themean number of spots was found for rolling two dice.What is the mean number of spots if three dice arerolled? 10.522. The formula for finding the variance for a probabilitydistribution iss 2 [(X m) 2 P(X)]Verify algebraically that this formula gives the sameresult as the shortcut formula shown in this section.23. Rolling a Die Roll a die 100 times. Compute the meanand standard deviation. How does the result compare withthe theoretical results of Example 5–5? Answers will vary.24. Rolling Two Dice Roll two dice 100 times and findthe mean, variance, and standard deviation of the sum ofthe spots. Compare the result with the theoretical resultsobtained in Exercise 20. Answers will vary.25. Extracurricular Activities Conduct a survey of thenumber of extracurricular activities your classmatesare enrolled in. Construct a probability distributionand find the mean, variance, and standarddeviation. Answers will vary.26. Promotional Campaign In a recent promotionalcampaign, a company offered these prizes and thecorresponding probabilities. Find the expected valueof winning. The tickets are free.Number of prizes Amount Probability11 $100,0001,000,00012 10,00050,00015 1,00010,000110 1001000If the winner has to mail in the winning ticket to claim theprize, what will be the expectation if the cost of the stampis considered? Use the current cost of a stamp for a firstclassletter. $1.56 with the cost of a stamp $0.445–18