PROBLEMS REQUIRED TO RETAKE EXPECTED VALUE, BINOMIAL AND GEOMETRIC PROBABILITY TEST

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1 PROBLEMS REQUIRED TO RETAKE EXPECTED VALUE, BINOMIAL AND GEOMETRIC PROBABILITY TEST I WILL NOT GRADE UNLESS PROBABILITY STATEMENTS AND CORRECT NOTATION ARE INCLUDED! I will be available for reassessments either on Wednesdays February 10 th or February 17 th (before or after school). You must turn in the following required problems by Monday the 8 th or the 15 th, respectively. However, please be aware that we will probably be testing over the next unit on Monday February 15 th. 1. Linda is a sales associate at a large auto dealership. At her commission rate of 25% of gross profit on each vehicle she sells, Linda expects to earn $350 for each car sold and $400 for each truck or SUV sold. She estimates her car sales on a sunny Saturday as follows: X=Cars sold Probability She estimates her truck or SUV sales on a sunny Saturday as follows: Y=Trucks/SUVs sold Probability Find the following: a. The mean and standard deviation of the number of cars sold. b. The mean and standard deviation of the number of trucks/suvs sold. c. The expected number of total vehicles sold. d. The standard deviation of the total vehicles sold. Assume independent. e. Calculate the mean and standard deviation for Linda s total commission on two sunny Saturdays. f. How many cars would you expect Linda to sell on TWO sunny Saturdays? What is the standard deviation? g. How many cars would you expect Linda to sell on ONE SUPER BUSY sunny Saturday when they have twice as much business? E(2x)? What is the standard deviation?

2 2. Vitamin D is essential for strong, healthy bones. Our bodies produce vitamin D naturally when sunlight falls upon the skin, or it can be taken as a dietary supplement. Although the bone disease rickets was largely eliminated in England during the 1950s, some people there are concerned that this generation of children is at increased risk because they are more likely to watch TV or play computer games that spend time outdoors. Recent research indicated that about 20% of British children are deficient in vitamin D. Suppose doctors test a group of elementary school children. a) What is the probability that the first vitamin D-deficient child is the 8 th one tested? b) How many kids do they expect to test before finding one who has this vitamin deficiency? c) If they test 320 children at this school, what is the probability that no more than 50 of them have the vitamin deficiency? 3. Rita is studying to be a real estate agent. About 61% of all people who take the licensing exam pass. a) What is the probability Rita needs three attempts to pass the exam? b) What is the probability Rita needs more than three attempts to pass the exam? 4. The director of a health club conducted a survey and found that 23% of the members used only the pool for workouts. What is the probability that for a random sample of 10 members, 4 used only the pool for workouts? 5. Natalie and Ryan are cross country runners for RHS. Each morning practice they run 3 miles on average about 19 and 18 minutes, respectively. The standard deviation of time is approximately 2 and 3 minutes, respectively. We will assume that they run independently. What is their expected difference in time to run three miles? And what is the standard deviation difference in time to run three miles? What is the probability that Natalie runs faster than Ryan? 6. Suppose the mean SAT verbal score is 525 with standard deviation 100, while the mean SAT math score is 575 with standard deviation of 100. What can be said about the mean and standard deviation of the combined math and verbal scores? Calculate each if possible.

3 7. Choose an American household at random and let the random variable X be the number of persons living in the household. If we ignore the few households with more than seven inhabitants, the probability distribution of X is as follows: Inhabitants Probability a) What is P X 5? b) What is X 5 P? c) What is P 2 X 4? d) What is X 1 P? 8. Our band members decide to wrap Christmas presents for folks as a fundraiser. Freshman band members collect present and donation for gift wrapping service with a mean time 4 minutes and standard deviation of 1 minute. The sophomores are responsible for wrapping the present with an average time of 12 minutes and standard deviation of 3 minutes. The juniors are in charge of tying ribbon and placing bows on gifts with a mean time of 2 minutes and standard deviation of 30 seconds. The seniors return wrapped gift to folks with an average time of 3 minutes and standard deviation of 1 minute. What is the mean and standard deviation of this assembly line approach to fundraising? What percent of the time does the total assembly line process take longer than 30 minutes? 9. About 8% of males are colorblind. A researcher needs 5 male subjects who are colorblind for an experiment and begins checking for potential subjects. What is the probability that the first colorblind male will be after the 6 th checked? 10. Suppose there are five marbles in a jar. They are identical except in color. Three of the marbles are red and two are blue. You are to draw out one marble, note its color, then replace it and draw out another marble. Let the random variable X represent the number blue marbles. Create a probability distribution model. Find the mean (expected value) and standard deviation for the number of blue marbles selected.

4 11. A computer repair shop has two work centers. The first center examines the computer to see what is wrong, and the second center repairs the computer. Let x 1 and x 2 be random variables representing the lengths of time in minutes to examine a computer (x 1) and to repair a computer (x 2). Assume x 1 and x 2 are independent random variables. Longterms history has shown the following times: Examine computer x 1: μ 1 = 28.1 minutes, Repair computer x 2: μ 2 = 90.5 minutes, σ 1 = 8.2 minutes σ 2 = 15.2 minutes a.) Let W = x 1 + x 2 be a random variable representing the total time to examine and repair the computer. Compute the mean, variance and standard deviation of W. b.) Suppose it costs $1.50 per minute to examine the computer and $2.75 per minute to repair the computer. Then Z = 1.50x x 2 is a random variable representing the service charges. Compute the mean, variance and standard deviation of Z. 12. In Chances: Risks and Odds in Everyday Life, James Burke claims that about 70% of all single men would welcome a woman taking the initiative in asking for a date. A random sample of 20 single men were asked if they would welcome a woman taking the initiative in asking for a date. What is the probability that a) at least 18 of the men will say yes? b) fewer than 3 of the men will say yes? c) none of the men will say yes? d) more than 5 of the men will say yes? e) exactly 10 of the men will say yes? f) 12 or more of the men will say yes? g) the first man to say yes is within the first 5 you ask? h) you must ask more than 4 before a man say yes? i) you ask less than 7 before a man says yes? j) what is the expected number of men to say yes? k) what is the standard deviation of the number of men to say yes? l) how many men do you expect to ask before the first one says yes?

5 13. The weight of medium-sized tomatoes selected at random from a bin at Kroger is a random variable with a mean μ = 10 ounces and a standard deviation σ =.5 ounce. a. Suppose we pick four tomatoes from the bin at random and put them in a bag. Let Y = the weight of the 4 tomatoes. Find the expected value (mean) and standard deviation of the random variable Y. b. What is the probability that the 4 randomly chosen tomatoes weigh less than 37.5 ounces? c. Suppose we pick two tomatoes at random from the bin. The difference in the weights is the random variable D = x 1 x 2. Find the mean and standard deviation (in ounces) of D. 14. Which of the following random variables is geometric? a) The number of times I have to roll a dice to get two 6s. b) The number of cards I deal from a well-shuffled deck of 52 cards until I get a heart. c) The number of digits I read in a randomly selected row on the random digits table until I find a 7. d) The number of 7s in a row of 40 random digits. e) The number of 6s I get if I roll a die 10 times. ON THE FOLLOWING MULTIPLE CHOICE QUESTIONS, YOU MUST SHOW WORK (PROBABILITY STATEMENTS, NOTATION AND CALCULATIONS) JUST CIRCLING A LETTER DOES NOT RECEIVE CREDIT. 1. A manufacturer makes light bulbs and claims that their reliability is 98 percent. Reliability is defined to be the proportion of non-defective items that are produced over the long term. If the company s claim is correct, what is the expected number of non-defective light bulbs in a random sample of 1,000 bulbs? a) 20 b) 200 c) 960 d) 980 e) 1,000

6 2. The XYZ Office Supplies Company sells calculators in bulk at wholesale prices, as well as individually at retail prices. Next year s sales depend on market conditions, but executives use probability to find estimates of sales for the coming year. The following tables are estimates for next year s sales. WHOLESALE SALES Number Sold 2,000 5,000 10,000 20,000 Probability RETAIL SALES Number Sold 600 1,000 1,500 Probability What profit does XYZ Office Supplies Company expect to make for the next year if the profit from each calculator sold is $20 at wholesale and $30 at retail? a) $10,590 b) $220,700 c) $264,750 d) $833,100 e) $1,002, The number of sweatshirts a vendor sells daily has the following probability distribution. Number of Sweatshirts x P(x) If each sweatshirt sells for $25, what is the expected daily total dollar amount taken in by the vendor from the sale of sweatshirts? a) $5.00 b) $7.60 c) $35.50 d) $38.00 e) $ Suppose that the distribution of a set of scores has a mean of 47 and a standard deviation of 14. If 4 is added to each score, what will be the mean and the standard deviation of the distribution of new scores? Mean Standard Deviation a) b) c) d) e) Suppose that for X = net amount won or lost in a lottery game, the expected value is E(X) = -$0.50. What is the correct interpretation of this value? a) The most likely outcome of a single play is a net loss of 50 cents. b) A player will have a net loss of 50 cents every single time he/she plays this lottery game. c) Over a large number of plays the average outcome for plays is a net loss of 50 cents. d) A mistake must have been made because it s impossible for an expected value to be negative. e) None of these

7 For 6 & 7: A fourth-grade teacher gives homework every night in both math and language arts. The time it takes her students to complete their math homework has a mean of 10 minutes and a standard deviation of 3 minutes. The time to complete the language arts assignment has a mean of 12 minutes and a standard deviation of 4 minutes. 6. The mean time to complete both homework assignments by her students is a) less than 22 minutes b) 22 minutes c) greater than 22 minutes d) cannot be determined since math and language arts completion times are dependent. e) None of these 7. The standard deviation to complete both homework assignments by her students is a) 16 minutes b) 9 minutes c) 5 minutes d) cannot be determined since math and language arts completion times are not independent. e) cannot be determined since math and language arts completion times are independent. 8. A test for extrasensory perception (ESP) involves asking a person to tell which of 5 shapes a circle, star, triangle, diamond, or heart appears on a hidden computer screen. On each trial, the computer is equally likely to select any of the 5 shapes. Suppose researchers are testing a person who does not have ESP and so is just guessing on each trial. What is the probability that the person guesses the first 4 shapes incorrectly but gets the fifth correct? a) 1 5 b) ( 4 5 )4 c) ( 4 5 )4 ( 1 5 )1 d) ( 5 1 ) (4 5 )4 ( 1 5 )1 e) 4 5 Continuation from question 8 write out the expanded form of the probability that the person guesses 4 correct out of 10 attempts (FROM FORMULA CHART AND NOTES). 9. Before planting a crop for the next year, a producer does a risk assessment. According to her assessment, she concludes that there are three possible net outcomes: a $7,000 gain, a $4,000 gain, or a $10,000 loss with probabilities 0.55, 0.20 and 0.25 respectively. The expected profit is: a) $3,850 b) $0 c) $2,150 d) $2,500 e) $800