Probability Exam 3 (chapter 4)

4.23 Approximately 10% of the glass bottles coming off a production line have serious defects in the glass. Two bottles are randomly selected for inspection. Find the expected value and the variance of the number of inspected bottles with serious defects.

E(x) = .2 Var(x) = .18

4.22 A fisherman is restricted to catching at most two red grouper per day when fishing in the Gulf of Mexico. A field agent for the wildlife commission often inspects the day's catch for boats as they come to shore near his base. He has found that the number of red grouper caught has the following distribution. Number of Grouper 0, 1, 2 Probability 0.2, 0.7, 0.1 Assuming that these records are representative of red grouper daily catches in the Gulf, find the expected value, the variance, and the standard deviation for the individual daily catch of red grouper.

E(x) = .9 Var(x) = .29 S(x) = .539

4.25 Two balanced coins are flipped. What are the expected value and variance of the number of heads observed?

E(x) = 1 Var(x) = .5

4.37 Four couples go to dinner together. The waiter seats the men randomly on one side of the table and the women randomly on the other side of the table. Find the expected value and variance of the number of couples who are seated across from each other.

E(x) = 1.001 Var(x) = 1.002999

4.73 Referring to Exercise 4.72, if each test costs $40, find the expected value and the variance of the total cost of conducting the tests to locate four positives. Is it highly likely that the cost of completing these tests will exceed $650?

E(x) = 10; Var(x) = 15 E(40x) = $400; Var(40x) = $24,000

5. A woman has eight keys on a key ring, one of which fits the door she wants to unlock. She randomly selects a key and tries it. If it does not unlock the door, she randomly selects another key from those remaining and tries to unlock the door with it. She continues in this manner until the door is unlocked. Let X be the number of keys she tries before unlocking the door, counting the key that actually worked. Find the expected number and variance of the number of keys she will try before she opens the door.

E(x) = 4.5 Var(x) = 5.25

3. At a large university the administration is studying the number of courses a student takes. Let X equal the number of courses for which a randomly selected student is registered. The probability mass function (pmf) is given as follows: X: 1 2 3 4 5 6 7 p(x): .01 .03 .13 .25 .39 .17 .02 Find the expected number and variance of the number of courses a student will take.

E(x) = 4.57 Var(x) = 1.2651

4.67 Suppose that 10% of the engines manufactured on a certain assembly line are defective. If engines are randomly selected one at a time and tested, find the probability that exactly two defective engines will be tested before a good engine is found.

.009

4.68 Referring to Exercise 4.67, given that the first two engines are defective, find the probability that at least two more defectives are tested before the first non-defective engine is found.

.01

5. Assume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the seventh component tested. Find the expected number and variance of the number of components tested before a defective component is found.

.017717 E(x) = 49 Var(x) = 2450

9. The Indiana Department of Transportation is concerned about the number of deer being struck by cars between Martinsville and Bloomington. They note the number of deer carcasses and other deer-related accidents over a 1-month period in a 2-mile interval. Based on the data collected they determine that the average number of deer struck by cars is 3.9 per 2-mile interval. What is the probability of zero deer strike incidents during any 2-mile interval between Martinsville and Bloomington?

.020242

4. Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55 percent of the 25,000 students do live within five miles of you. You randomly contact students from the college until one says he or she lives within five miles of you. What is the probability that you need to contact four people?

.050119

4.72 The employees of a firm that does asbestos cleanup are being tested for indications of asbestos in their lungs. The firm is asked to send four employees who have positive indications of asbestos on to a medical center for further testing. If 40% of the employees have positive indications of asbestos in their lungs, find the probability that six employees who do not have asbestos in their lungs must be tested before finding the four who do have asbestos in their lungs.

.100329

1. A small voting district has 101 female voters and 95 male voters. A random sample of 10 voters is drawn. What is the probability exactly 7 of the voters will be female?

.130396

4. Suppose we know that births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability that we observe 5 births in a given 2 hour interval?

.13768

11. Suppose that there are ten cars available for you to test drive, and five of the cars have turbo engines. If you test drive three of the cars, what is the probability that two of the three cars that you drive will have turbo engines?

.416667

7. A wallet contains 3 $100 bills and 5 $1 bills. You randomly choose 4 bills. What is the probability that you will choose exactly 2 $100 bills?

.428571

3. Suppose there is a disease, whose average incidence is 2 per million people. What is the probability that a city of 1 million people has 4 or fewer cases of the disease?

.947347

13. Suppose you run a charter school and you admit students through a lottery system. You have 734 applicants, of which 321 are boys and 413 girls. You are only admitting the first 150 students that are randomly selected. Considering your school has more girls than boys, you hope to get more boys this year. What is the probability of admitting 90 boys to your school?

[(321C90)*(413C60)]/(734C150)

5. Seven television tubes are chosen at random from a shipment of 240 television tubes of which 15 are defective. a. What is the probability that 4 of the chosen televisions are defective? b. What is the probability 5 of the chosen televisions are defective? c. What is the probability that at most 5 of the chosen televisions are defective? d. What is the probability that at least 1 of the chosen televisions are defective? e. What is the expected number of defective TVs? f. If it costs $75 to repair a defective television, what is the expected total repair cost?

a .000307 b .000009 c .99999986405561 d .367272 e .4375 f $32.8125

8. Vehicles pass through a junction on a busy road at an average rate of 300 per hour. a. Find the probability that none passes in a given minute. b. What is the expected number passing in two minutes? c. Find the probability that this expected number actually pass through in a given two-minute period.

a .006738 b. 10 c .12511

6. 10% of applicants for a job possess the right skills. A company interviews applicants one at a time until they find a qualified applicant a. What is the probability that they will interview exactly ten applicants? b. What is the probability that they will interview at least ten applicants?

a .038742 b .38742

4.74 People with O− blood are called universal donors because they may give blood to anyone without risking incompatibility due blood type factors (A and B) or to the Rh factor. Of the persons donating blood at a clinic, 9% have O− blood. Find the probabilities of the following events. a The first O− donor is found after blood typing five people who were not O−. b The second O− donor is the sixth donor of the day.

a .056163 b .027773

2. Births in a hospital occur randomly at an average rate of 1.8 births per hour. a. What is the probability of observing 4 births in a given hour at the hospital? b. What about the probability of observing at least 2 births in a given hour at the hospital?

a .072302 b .537163

4.54 A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has four identical components, each with a probability of 0.15 of failing in less than 1000 hours. The subsystem will operate if any two or more of the four components are operating. Assuming that the components operate independently, find the probabilities of the following events. a Exactly two of the four components last longer than 1000 hours. b The subsystem operates for longer than 1000 hours.

a .097538 b .988019

4.75 A geological study indicates that an exploratory oil well drilled in a certain region should strike oil with probability 0.25. Find the probabilities of the following events. a The first strike of oil comes after drilling three dry (nonproductive) wells. b Three dry wells are hit before obtaining the third strike of oil. c What assumptions must be true for your answers to be correct?

a .105469 b .065918 c. each drill must be independent

4.46 When testing insecticides, the amount of the chemical when given all at once will result in the death of 50% of the population is called the LD50, where LD stands for lethal dose. If 40 insects are placed in separate Petri dishes and treated with an insecticide dosage of LD50, find the probabilities of the following events. a Exactly 20 survive b At most 15 survive c At least 20 survive d Does it matter whether the insects are placed in separate Petri dishes as they were here or in one large Petri dish? Justify your answer. e Find the number expected to survive out of 40. f Find the variance and standard deviation of the number of survivors out of 40.

a .125371 b .07693 c .562685 d need separate dishes to be independent e E(x) = 20 f Var(x) = 10; S(x) = 3.1623

12. Customer arrivals at a checkout counter in a department store with an average of seven per hour. For a given hour, find the probabilities of the following events. a. Exactly seven customers arrive. b. No more than two customers arrive. c. At least two customers arrive.

a .149003 b .029636 c .992705

4.66 Let X denote a negative binomial random variable, with p = 0.6. Find P(X ≥ 3) for the following values of r. a r = 2 b r = 4

a .1792 b .4557

4.83 John and Fred agree to play a series of tennis games. The first one to win three games is declared the overall winner. Suppose that John is a stronger tennis player than Fred. So the probability that John wins each game is 0.6, and the outcome of each game is independent of the outcomes of the other games. a Find the probability that John wins the series in i games, for i = 3, 4, 5. b Compare the probability that John wins with the probability that he would win if they played a win two-out-of-three series.

a .216; .2592; .20736 b .648 > .6

4.52 The efficacy of the mumps vaccine is about 80%; that is, 80% of those receiving the mumps vaccine will not contract the disease when exposed. Assume that each person's response to the mumps is independent of another person's response. Find the probability that at least one exposed person will get the mumps if n are exposed where: a n = 2. b n = 4.

a .36 b .5904

4.38 A family has an old watchdog that is hard of hearing. If a burglar comes, the probability that the dog will hear him is 0.6. If the dog does hear the burglar, the probability that he will bark and awaken the family is 0.8. Suppose burglars come to this family's house on three separate occasions. Let X be the number of times (out of the three attempted burglaries) the dog barks and awakens the family. Find the mean and standard deviation of X.

mean: E(x) = 1.44 standard deviation: sqrt[Var(x)] = .865332

4.78 An interviewer is given a list of potential people she can interview. Suppose that the interviewer needs to interview five people and that each person independently agrees to be interviewed with probability 0.6. Let X be the number of people she must ask to be interviewed to obtain her necessary number of interviews. a What is the probability that she will be able to obtain the five people by asking no more than seven people? b What is the expected value and variance of the number of people she must ask to interview five people?

a .419904 b. E(x) = 8.33; Var(x) = 5.56

4.45 A machine that fills milk cartons underfills a certain proportion p. If 50 cartons are randomly selected from the output of this machine, find the probability that no more than 2 cartons are underfilled when: a. p = 0.05. b. p = 0.1.

a .540533 b .111729

4.47 Among persons donating blood to a clinic, 85% have Rh+ blood (that is, the Rhesus factor is present in their blood.) Six people donate blood at the clinic on a particular day. a Find the probability that at least one of the five does not have the Rh factor. b Find the probability that at most four of the six have Rh+ blood. c The clinic needs six Rh+ donors on a certain day. How many people must donate blood to have the probability of obtaining blood from at least six Rh+ donors over 0.95?

a .622839 b .223516 c n = 9

4.65 Let X denote a random variable that has a geometric distribution with a probability of success on any trial denoted by p. Let p = 0.1. a Find P(X ≥ 2). b Find P(X ≥ 4|X ≥ 2)

a .81 b .81

4.24 Two construction contracts are to be randomly assigned to one or more of three firms—I, II, and III. A firm may receive more than one contract. Each contract has a potential profit of $90,000. a Find the expected potential profit for firm I. b Find the expected potential profit for firms I and II together.

a. $60,000 b. $120,000

10. Lot acceptance sampling procedures for an electronics manufacturing firm call for sampling n items from a lot of N items and accepting the lot if X <= c, where X is the number of nonconforming items in the sample. For an incoming lot of 20 transistors, 5 are to be sampled. Find the probability of accepting the lot if c = 1 and the actual number of nonconforming transistors in the lot are as follows: a. 0 b. 1 c. 2 d. 3

a. 1 b. 1 c .947368 d .859649

4.26 In a promotional effort, new customers are encouraged to enter an online sweepstakes. To play, the new customer picks 9 numbers between 1 and 50, inclusive. At the end of the promotional period, 9 numbers from 1 to 50, inclusive, are drawn without replacement from a hopper. If the customer's 9 numbers match all of those drawn (without concern for order), the customer wins $5,000,000. a What is the probability that a randomly selected new customer wins the $5,000,000? b What is the expected value and variance of the winnings? c If a new customer had to mail in the picked numbers, assuming that the cost of postage and handling is $0.50, what is the expected value and variance of the winnings?

a. 1 / (50 choose 9) b. E(x) = $0.0019956625; Var(x) = $9978.3123337183 c. E(x) = $-0.498004; Var(x) = $9978.3083503588

4.17 You are to pay $1 to play a game that consists of drawing one ticket at random from a box of numbered tickets. You win the amount (in dollars) of the number on the ticket you draw. The following two boxes of numbered tickets are available. I. [0, 1, 2] II. [0, 0, 0 1, 4] a Find the expected value and variance of your net gain per play with box I. b Repeat part (a) for box II. c Given that you have decided to play, which box would you choose and why?

a. E(x) = $0; Var(x) = $0.67 b. E(x) = $0; Var(x) = $2.40

4.70 Referring to Exercise 4.67, find the mean and the variance of the number of defectives tested before the following events occur. a The first non-defective engine is found. b The third non-defective engine is found.

a. E(x) = .111; Var(x) = .123 b. E(x) = .333; Var(x) = .370

4. Based on long years of experience, a supervisor in a large factory has determined the probability distribution of X, the number of employees that call in sick on Monday is as follows: X: 0 1 2 3 4 5 6 7 p(x): .005 .025 .310 .340 .220 .080 .019 .001 a. How many people can the manager expected to be out on Monday? b. Find the variance and standard deviation of the number of people who call in sick on Monday.

a. E(x) = 3.066 b. Var(x) = 1.7764 S(x) = 1.08519

4.53 Refer to Exercise 4.52. a How many vaccinated people must be exposed to the mumps before the probability that at least one person will contract the disease is at least 0.95? b In 2006, an outbreak of mumps in Iowa resulted in 605 suspected, probable, and confirmed cases. Given broad exposure in this state of 2.9 million people, do you find this number to be excessively large? Justify your answer.

a. n = 14 b. E(x) = 580,000 so 605 is small

4.58 From a large lot of memory chips for use in personal computers, n are to be sampled by a potential buyer, and the number of defectives X is to be observed. If at least one defective is observed in the sample of n, the entire lot is to be rejected. Find n so that the probability of detecting at least one defective is approximately 0.95 if the following percentages are correct. a Of the lot of memory chips, 10% are defective. b Of the lot of memory chips, 5% are defective.

a. n = 28 b. n = 58

6. A crate contains 50 light bulbs of which 5 are defective and 45 are not. A Quality Control Inspector randomly samples 4 bulbs without replacement. Let X the number of defective bulbs selected. a. Find the probability mass function, p(x) b. Find the expected number of defective light bulbs.

a. see hw 14 b .4

2. Among 15 applicants for an open position, 9 are women and 6 are men. Suppose that four applicants are randomly selected from the applicant pool for final interviews. Let X be the number of female applicants among the final four. a. Find the probability function for X. b. Graph the probability function of X

see hw 10

3. Daily sales records for a cabin cruiser dealership show that it will sell 0, 1, 2, or 3 cabin cruisers, with probabilities as listed: Number of Sales 0 1 2 3 Probability 0.6 0.15 0.15 0.10 a. Find the probability distribution for X, the number of sales in a 2-day period, assuming that the sales are independent from day to day. b. Find the probability that two or more sales are made in the next 2 days.

see hw 10

4. A woman has eight keys on a key ring, one of which fits the door she wants to unlock. She randomly selects a key and tries it. If it does not unlock the door, she randomly selects another key from those remaining and tries to unlock the door with it. She continues in this manner until the door is unlocked. Let X be the number of keys she tries before unlocking the door, counting the key that actually worked. Find the probability function of X.

see hw 10

4.1 Circuit boards from two assembly lines set up to produce identical boards are mixed in one storage tray. As inspectors examine the boards, they find that it is difficult to determine whether a board comes from line A or line B. A probabilistic assessment of this question is often helpful. Suppose that the storage tray contains 10 circuit boards of which six came from line A and four from line B. An inspector selects two of these identical-looking boards for inspection. He is interested in X, the number of inspected boards from line A. a Find the probability function for X. b Graph the probability function of X. c Find the distribution function of X. d Graph the distribution function of X.

see hw 10

4.10 When turned on, each of the three switches in the accompanying diagram works properly with probability 0.9. If a switch is working properly, current can flow through it when it is turned on. Find the probability distribution for X, which is the number of closed paths from s to t, when all three switches are turned on. *see textbook for image*

see hw 10

4.11 At a miniature golf course, players record the strokes required to make each hole. If the ball is not in the hole after five strokes, the player is to pick up the ball and record six strokes. The owner is concerned about the flow of players at hole 7. (She thinks that players tend to get backed up at that hole.) She has determined that the distribution function of X, the number of strokes a player takes to complete hole 7 to be the following: F(x) = 0, x < 1; 0.05, 1 ≤ x < 2; 0.15, 2 ≤ x < 3; 0.35, 3 ≤ x < 4; 0.65, 4 ≤ x < 5; 0.85, 5 ≤ x < 6; 1, x ≥ 6 a Graph the distribution function of X. b Find the probability function of X. c Graph the probability function of X. d Based on (a) through (c), are the owner's concerns substantiated? Justify your answer.

see hw 10

4.12 Observers have noticed that the distribution function of X, which is the number of commercial vehicles that cross a certain toll bridge during a minute, is as follows: F(x) = 0, x < 0; 0.20, 0 ≤ x < 1; 0.50, 1 ≤ x < 2; 0.85, 2 ≤ x < 4; 1, x ≥ 4 a Graph the distribution function of X. b Find the probability function of X. c Graph the probability function of X.

see hw 10

4.2 Among 12 applicants for an open position, 7 are women and 5 are men. Suppose that three applicants are randomly selected from the applicant pool for final interviews. Let X be the number of female applicants among the final three. a Find the probability function for X. b Graph the probability function of X. c Find the distribution function of X. d Graph the distribution function of X.

see hw 10

4.3 The median annual income for heads of households in a certain city is $44,000. Four such heads of household are randomly selected for inclusion in an opinion poll. Let X be the number (out of the four) who have annual incomes below $44,000. a Find the probability distribution of X. b Graph the probability distribution of X. c Find the distribution function of X. d Graph the distribution function of X. e Is it unusual to see all four below $44,000 in this type of poll? (What is the probability of this event?)

see hw 10

4.4 In 2005, Derrek Lee led the National Baseball League with a 0.335 batting average, meaning that he got a hit on 33.5% of his official times at bat. In a typical game, he had three official at bats. a Find the probability distribution for X, the number of hits Lee got in a typical game. b What assumptions are involved in the answer? Are the assumptions reasonable? c Is it unusual for a good hitter to go 0 for 3 in one game?

see hw 10

4.5 A commercial building has four entrances, numbered I, II, III, and IV. Three people enter the building at 9:00 a.m. Let X denote the number of people who select entrance I. Assuming that the people choose entrances independently and at random, find the probability distribution for X . Were any additional assumptions necessary for your answer?

see hw 10

4.8 Four microchips, two of which are defective, are to be placed in a computer. Two of the four chips are randomly selected for inspection before the computer is assembled. Let X denote the number of defective chips found among the two inspected. a Find the probability distribution for X. b Find the probability that no more than one of the two inspected chips was defective.

see hw 10

4.9 Of the people who enter a blood bank to donate blood, 1 in 3 has type O+ blood, and 1 in 20 has type O− blood. For the next three people entering the blood bank, let X denote the number with O+ blood, and let Y denote the number with O− blood. Assume the independence among the people with respect to blood type. a Find the probability distributions of X and of Y. b Find the probability distribution of X + Y , which is the number of people with type O blood.

see hw 10

5. At a miniature golf course, players record the strokes required to make each hole. If the ball is not in the hole after five strokes, the player is to pick up the ball and record six strokes. The owner is concerned about the flow of players at hole 7. (She thinks that players tend to get backed up at that hole.) She has determined that the distribution function of X, the number of strokes a player takes to complete hole 7 to be the following: F(x) = *see hw 10* a. Graph the cumulative distribution function of X. b. Find the probability mass function of X. c. Graph the probability mass function of X. d. Based on (a.) through (c.), are the owner's concerns substantiated? Justify your answer.

see hw 10

2. Let m be a constant and ... *see hw 11*

see hw 11

2. *see hw 12*

see hw 12

3. For the Bernoulli distribution verify: *see hw 12*

see hw 12

3. For the Geometric Distribution verify *see hw 13*

see hw 13

4.33 A merchant stocks a certain perishable item. He knows that on any given day he will have a demand for two, three, or four of these items, with probabilities 0.2, 0.3, and 0.5, respectively. He buys the items for $1 each and sells them for $1.20 each. Any items left at the end of the day represent a total loss. How many items should the merchant stock to maximize his expected daily profit?

stock 2 to maximize profit