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25-2 Which of the following pairs of events are mutually exclusive?

27-1 these two events are said to be:

34-1 300 randomly selected individuals are studied with results

37-2 The law of large numbers makes it possible

38-2 Define the complement to the event A

40-1 Chance Process

44-2 Which of the following is NOT a possible probability

71-1 estimate from the above table

533-2 "Among twenty-five articles, nine are defective,"

534-1 The checking accounts of Save-More Bank are categorized by

536-1 The depositors at Save-More Bank are categorized by age and sex.

538-1 P (Packers score a first down)

539-1 400 adult males with angina pectoris are classified by age and weight

540-1 "Each member of a sample of 3,750 women aged 30-39 was measured for"

541-1 exposure and disease in a group of 1000 people

542-1 An epidemiologist feels that railroads have something to do with the

545-1 "Among twenty-five articles eight are defective, six having only"

546-1 "The probability of selecting a lower division student, given the"

549-1 "person selected is a plumber, the probability that"

550-1 Which of the following statements is appropriate for describing the 40%

553-2 "If this die is thrown and the top face shows an odd number,"

558-1 "If it lands heads, a marble is drawn from Urn H"

560-1 what is your estimate of the probability that:

1885-1 How may the standard deviation be expressed?

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Q: Which of the following pairs of events are mutually exclusive? a. A: the odd numbers; B: the number 5 b. A: the even numbers; B: the numbers greater than 10 c. A: the numbers less than 5; B: all negative numbers d. A: the numbers above 100; B: the numbers less than -200 e. A: negative numbers; B: odd numbers

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Q: One card is drawn from a standard 52 card deck. In describing the occurrence of two possible events, an Ace and a King, these two events are said to be: (a) independent (b) mutually exclusive (c) random variables (d) randomly independent.

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Q: Suppose a certain opthalmic trait is associated with eye color. 300 randomly selected individuals are studied with results as follows: EYE COLOR TRAIT | Blue | Brown | Other | Total ____________________________________ Yes | 70 | 30 | 20 | 120 ____________________________________ No | 20 | 110 | 50 | 180 ____________________________________ Total | 90 | 140 | 70 | 300 A. What is the probability that a person has blue eyes? B. What would you expect to be the value P(having the trait and blue eyes) if eye color and trait status were independent? C. Which of the following expressions describes the relation- ship between the events A = a person has brown eyes and B = a person has blue eyes? (circle the correct answer) i. independent ii. exhaustive iii. simple iv. mutually exclusive

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Q: True or False? If False, correct it. The law of large numbers makes it possible to predict long run relative frequencies but not particular chance events.

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Q: Define the complement to the event A.

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Q: Define the following term and give an example of its use. Your example should not be one given in class or in a handout. Chance Process

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Q: Which of the following is NOT a possible probability? a. 25/100 b. 1.25 c. 1 d. 0

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Q: A sample of 1000 persons screened for a certain disease is distributed according to height and disease status resulting from a clinical exam as follows: DISEASE STATUS None Mild Moderate Severe Totals |-------|---------|------------|----------| Tall | 122 | 78 | 139 | 61 | 400 |-------|---------|------------|----------| HEIGHT Medium | 74 | 51 | 90 | 35 | 250 |-------|---------|------------|----------| Short | 104 | 71 | 121 | 54 | 350 |-------|---------|------------|----------| 300 200 350 150 1000 What would you estimate from the above table to be the probability of being medium or short in height and having moderate or severe disease status? a. 600/1000 * 500/1000 d. 300/600 b. 300/500 e. 800/1000 c. 300/1000

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Q: Among twenty-five articles, nine are defective, six having only minor defects and three having major defects. Determine the probability that an article selected at random has major defects given that it has defects. a. 1/3 b. .25 c. .24 d. .08

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Q: The checking accounts of Save-More Bank are categorized by age of account and balance in account. We are going to select an account at random from this group of 2000 accounts. Balance | | | Age of Account | $0 - $99 | $100-$499 | $500 or more --------------------------------------------------------------- less than 3 years | 700 | 100 | 400 3 or more years | 200 | 400 | 200 --------------------------------------------------------------- i) Then P($500 or more0 - 3 years) = a) 6/7 b) 2/3 c) 1/3 d) 1/5 e) none of these ii) Then P[($0 - $99) or (3 years or more)] = a) 9/10 b) 9/20 c) 3/4 d) 2/5 e) none of these iii) Then P($100 or more) = a) 1/4 b) 3/20 c) 11/20 d) 3/10 e) none of these iv) What is the conditional probability that the account has a balance under $100, given that it is less than 3 years old? a) 7/9 b) 9/20 c) 3/4 d) 7/12 e) none of these v) Are age of account and balance in account independent at Save-More Bank? Why or why not? vi) Suppose fourteen accounts are drawn at random from this bank. Let G be the event: "At least five accounts are less than 3 years old". State G', the complement of G.

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Q: The depositors at Save-More Bank are categorized by age and sex. We are going to select an individual at random from this group of 2000 depositors. Sex Age | Male | Female ------------------------------------- 30 or less | 800 | 600 31 or more | 400 | 200 ------------------------------------- i) Then P(Female30 or less) = a) 2/5 b) 3/4 c) 3/7 d) 3/10 e) none of these ii) Then P[Male or (31 or more)] = a) 1/5 b) 3/10 c) 1/2 d) 7/10 e) none of these iii) Then P(Female) = a) 3/10 b) 2/5 c) 3/5 d) 2/3 e) none of these iv) What is the conditional probability that the depositor drawn is 30 or less, given that he is a male? a) 2/3 b) 7/10 c) 4/7 d) 2/5 e) none of these v) Are age of depositor and sex of depositor independent at Save-More Bank? Why or why not?

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Q: The Green Bay Packers and Chicago Bears are playing their annual charity game. It is third down, four yards to go for the Packers on their own 45 yard line. Based on previous performances by the Packers, the next play will see one of the following occurrences, with probabilities as indicated: P (pass incomplete) = .25 P (pass complete for first down) = .15 P (pass complete, no first down) = .05 P (interception) = .03 P (run for first down) = .20 P (busted play, quarterback sacked) = .25 P (whistle is blown before play) = .05 Find the following probabilities for the next play: a) P (Packers score a first down) b) P (pass play is tried) c) P (Packers score a first downpass play is tried)

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Q: 400 adult males with angina pectoris are classified by age and weight as follows: | Weight in Pounds | Age (years) | 130-149 | 150-169 | 170-189 | >=190 | Total ----------------------------------------------------------- 30-39 | 10 | 20 | 20 | 40 | 90 40-49 | 10 | 15 | 50 | 70 | 145 50-59 | 5 | 15 | 50 | 40 | 110 60-69 | 5 | 10 | 15 | 25 | 55 ----------------------------------------------------------- | 30 | 60 | 135 | 175 | 400 Using the table, find for a randomly selected individual from this population the probability that he or she: a) Is in the age interval 40-49 b) Is in the age interval 40-49 and weighs 170-189 lbs c) Is in the age interval 40-49 or 60-69 d) Is in the age interval 40-49 or 60-69 and weighs 150-169 lbs e) Is in the age interval 40-49 given a weight between 150-169 lbs f) Weighs less than 170 lbs g) Weighs less than 170 lbs and is less than 50 years h) Weighs less than 170 lbs given that he is less than 50 years

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Q: Each member of a sample of 3,750 women aged 30-39 was measured for unaided distance vision in both right and left eyes. The results are presented in the following table: | LEFT EYE | RIGHT EYE | Highest Second Third Lowest | Total | grade grade grade grade | --------------------------------------------------------- Highest grade | 750 130 60 35 | 975 Second grade | 125 775 210 40 | 1150 Third grade | 60 180 885 100 | 1225 Lowest grade | 20 40 90 250 | 400 --------------------------------------------------------- Total | 955 1125 1245 425 | 3750 For a randomly selected person from the population sampled above, what is your estimate of the probability that: a) the left eye will fall into the 3rd grade of unaided distance vision b) the left eye will have the highest grade given that the right eye has the lowest grade c) the right eye will have the highest grade and the left eye will have the lowest grade

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Q: The following two-way table shows the frequencies of occurrence of a hypothetical exposure and disease in a group of 1000 people. Disease Present Absent | Present 75 325 | 400 Exposure Absent 25 575 | 600 --------------------------- 100 900 1000 a. What is the probability of exposure in the group? b. What is the joint probability of both exposure and disease being present in the group? c. Compute the probability of disease being present conditional on the presence of exposure and conditional on the absence of exposure.

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Q: An epidemiologist feels that railroads have something to do with the development of a new disease because the probability of a person's living within a mile of railroad tracks, given that he has the disease, is .80. Do you agree with him? Why or why not?

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Q: Among twenty-five articles eight are defective, six having only minor defects and two having major defects. Determine the pro- bability that an article selected at random has major defects given that it has defects. (a) .08 (c) 1/3 (b) .25 (d) .24

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Q: A dormitory on campus houses 200 students. 120 are male, 50 are upper division students, and 40 are upper division male students. A student is selected at random. The probability of selecting a lower division student, given the student is a female, is: (a) 7/8 (d) 7/20 (b) 7/15 (e) 1/4 (c) 2/5

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Q: A local trade union consists of plumbers and electricians. Classified according to rank: Apprentice Journeyman Master ------------------------------------ Plumbers 25 | 20 | 30 75 ------------------------------------ Electricians 15 | 40 | 20 75 ------------------------------------- 40 60 50 A member of the union is selected at random. Given that the person selected is a plumber, the probability that he is a journeyman is: a. 1/2 b. 1/3 c. 4/15 d. 2/15 e. none of these.

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Q: In an effort to get at the source of an outbreak of Legionnaire's disease at the 197? APHA convention, a team of medical detectives (i.e. epidemiologists) carried out a case-control study involving all 50 cases and a sample of 200 non-cases out of the 4000 persons attending the convention. Among the results, it was found that 40% of the cases went to a cocktail party given by a large drug company on the second night of the convention, whereas 10% of the controls attended the same party. Which of the following statements is appropriate for describing the 40% of cases who went to the party? (C = case, P = attended party) a. Pr(C|P) = .40 b. Pr(P|C') = .40 c. Pr(C|P') = .40 d. Pr(P'|C) = .40 e. none of these

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Q: Suppose a loaded die has the following model: Face 1 2 3 4 5 6 Probability 0.3 0.1 0.1 0.1 0.1 0.3 If this die is thrown and the top face shows an odd number, a. What is the probability that the die shows a four? b. What is the probability that the die shows a 1?

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Q: There are two urns marked H and T. Urn H contains 2 red marbles and 1 blue marble. Urn T contains 1 red and 2 blue marbles. A coin is to be tossed. If it lands heads, a marble is drawn from Urn H. If it lands tails a marble is drawn from Urn T. Find the following pro- babilities: ------------------- | H | T | ----------------------------- R | 2/6 | 1/6 | 1/2 ----------------------------- B | 1/6 | 2/6 | 1/2 ----------------------------- | 1/2 | 1/2 | 1 ------------------------- a. P(heads and red) b. P(tails) c. P(red) d. P(blue) e. P(heads|red)

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Q: A sample of 2000 individuals is distributed according to eye color and the presence or absence of a certain opthalmic trait as follows: | EYE COLOR | TRAIT | Blue Brown Other | ---------------------------------------------- Yes | 400 270 130 | 800 No | 200 650 350 | 1200 ---------------------------------------------- Total | 600 920 480 | 2000 In a random selection of an individual from the study population, what is your estimate of the probability that: a. the person has blue eyes? _______________ b. the trait is present and the person has brown eyes? ____________ c. the person has neither brown nor blue eyes given that the trait is absent? _______________ d. the person has neither brown nor blue eyes and the trait is pre- sent? _______________ e. the person does not have brown eyes? _______________ f. the person has blue eyes or has neither blue nor brown eyes? _____________ g. the person does not have the trait or does not have brown eyes? _______________

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Q: How may the standard deviation be expressed? a. A point on a z-score scale. b. A distance on a z-score scale. c. An index on a squared numerical scale. d. (a) and (c) are both correct. e. (b) and (c) are both correct.

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A: d. A: the numbers above 100; B: the numbers less than -200

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A: (b) mutually exclusive.

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A: A. 90/300 = .3 B. (120/300)*(90/300) = .12 C. iv. mutually exhaustive

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A: True.

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A: The complement of A is the set of elements which do not belong to A.

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A: Definition: A process producing results or outcomes where the next outcome cannot be specified in advance, but where the long term chances of that outcome can be determined. Example: If we were to regard dropping a baited line into water suspected of harboring flounder as a random process, we would be taking the position that it's not possible to accurately forecast if the next dropping of a line will return a fish, but that if we repeatedly drop a properly baited line it will result in a catch some percentage of the time.

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A: b. 1.25 0 <= Probability <= 1.0

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A: c. 300/1000 (90 + 35 + 121 + 54)/1000 = 300/1000

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A: a. 1/3 P[MD/D] = (3/25)/(9/25) = 3/9 = 1/3

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A: i) c) 1/3 P($500 or more0 - 3 years) = 400/1200 = 1/3 ii) c) 3/4 P(($0 - $99) or (3 years or more)) = P($0 - $99) + P(3 years or more) - P(($0 - $99) and (3 years)) = 900/2000 + 800/2000 - 200/2000 = 3/4 iii) c) 11/20 P($100 or more) = P($100 - $499) + P($500 or more) = 500/2000 + 600/2000 = 11/20 iv) d) 7/12 P($0 - $990 - 3 years) = 700/1200 = 7/12 v) The two variables are not independent because P(0 - 3 years$0 - $99) =/= P(0 - 3 years$100 - $499) =/= P(0 - 3 years$500 or more) vi) G': "At least 10 accounts are three years old or older" or "less than five accounts are less than three years old"

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A: i) c) 3/7 P(Female30 or less) = 600/1400 = 3/7 ii) d) 7/10 P[Male or (31 or more)] = P(Male) + P(31 or more) - P(Male and 31 or more) = (1200/2000) + (600/2000) - (400/2000) = 1400/2000 = 7/10 iii) b) 2/5 P(Female) = 800/2000 = 2/5 iv) a) 2/3 P(30 or lessMale) = 800/1200 = 2/3 v) They are not independent because P(30 or lessMale) =/= P(30 or lessFemale)

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A: a) .15 + .20 = .35 b) .25 + .15 + .05 + .03 = .48 c) .15/.48 = .3125

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A: a) .3625 = 145/400 b) .1250 = 50/400 c) .5000 = 145/400 + 55/400 d) .0625 = 15/400 + 10/400 e) .2500 = 15/60 f) .2250 = 30/400 + 60/400 g) .1375 = 10/400 + 10/400 + 20/400 + 15/400 h) .2340 = (10+10+20+15)/(145+90)

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A: a) 1245/3750 = .3320 b) 20/400 = .05 c) 35/3750 = .0093

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A: a. P(exposure) = 400/1000 = .40 b. P(exposure and disease) = 75/1000 = .075 c. P(diseaseexposure present) = 75/400 = .1875 P(diseaseexposure absent) = 25/600 = .0417

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A: You don't know whether P(living near tracksdisease) is equal to P(living near tracks), therefore you cannot evaluate whether the two events are independent or not.

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A: (b) .25 P(MDD) = P(MD and D)/P(D) = (2/25)/(8/25) = .25

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A: (a) 7/8 Given the above information, one can construct the following table: Lower Upper Male | 80 | 40 | 120 ------------------------------------- Female | 70 | 10 | 80 ------------------------------------- | 150 | 50 | 200 P(LowerFemale) = 70/80

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A: c. 4/15 20/75 = 4/15

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A: e. none of these Pr(PC) = .40 is the correct notation.

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A: a. P(4|odd number) = 0 b. P(1|odd number) = P(1 and odd number)/P(odd number) = .3/(.3 + .1 + .1) = .6

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A: (a) P(heads, red) = (1/2)*(2/3) = 1/3 (b) P(tails) = 1/2 (c) P(red) = ((1/2)*(2/3) + (1/2)*(1/3)) = 1/2 (d) P(blue) = ((1/2)*(1/3) + (1/2)*(2/3)) = 1/2 (e) P(heads|red) = P(heads intersect red)/P(red) = (1/3)/(1/2) = (1/3)*(2/1) = 2/3

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A: a. 600/2000 = 3/10 b. 270/2000 = 27/200 c. 350/1200 = 7/24 d. 130/2000 = 13/200 e. (600 + 480)/2000 = 27/50 f. P(Blue union Other) = P(Blue) + P(Other) - P(Blue and Other) = 600/2000 + 480/2000 - 0 = 1080/2000 = 27/50 g. P(No trait union (Blue or Other)) = 1200/2000 + 1080/2000 - 550/2000 = 1730/2000 = 173/200

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A: b. A distance on a z-score scale.

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Item is still being reviewed Multiple Choice BASICTERMS/PROB PROBABILITY T= 2 Comprehension D= 2 General

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Item is still being reviewed Multiple Choice BASICTERMS/PROB EVENTS PROBABILITY T= 2 Comprehension D= 2 General

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Based upon item submitted by D. Kleinbaum - Univ of North Carolina Numerical Answer INDEPENDENT OTHER/EVENTS BASICTERMS/PROB SIMPLEPROBABILIT EVENTS PROBABILITY T=10 Comprehension D= 3 General Biological Sciences ***Multiple Parts***

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Based upon item submitted by W. J. Hall - Univ. of Rochester True/False BASICTERMS/PROB PROBABILITY T= 2 Comprehension D= 3 General

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Based upon item submitted by S. Selvin - UC Berkeley Definition OTHER/EVENTS BASICTERMS/PROB EVENTS PROBABILITY T= 2 Comprehension D= 1 General

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Based upon item submitted by J. Warren - UNH Definition BASICTERMS/PROB EVENTS I650I PROBABILITY T= 5 Comprehension D= 2 General

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Item is still being reviewed Multiple Choice SIMPLEPROBABILIT PROBMODELS PROBABILITY T= 2 Comprehension D= 1 General Education

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Item is still being reviewed Multiple Choice JOINTDISTRIBUTIO CONTINGENCYTABLE PROBDISTRIBUTION PROBABILITY CHISQUARE NONPARAMETRIC STATISTICS T= 2 Computation D= 3 Biological Sciences General

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Based upon item submitted by H. B. Christensen - BYU Multiple Choice CONDITIONALPROBA NONBAYESIAN PROBABILITY T= 2 Computation Application D= 4 General Business

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Item is still being reviewed Multiple Choice INDEPENDENT CONDITIONALPROBA JOINTPROBABILITY EVENTS PROBABILITY T=10 Computation Comprehension D= 3 General ***Multiple Parts***

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Item is still being reviewed Multiple Choice INDEPENDENT CONDITIONALPROBA JOINTPROBABILITY EVENTS PROBABILITY T=10 Computation Comprehension D= 4 General

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Based upon item submitted by R. H. Locher - Marquette Numerical Answer SIMPLEPROBABILIT CONDITIONALPROBA PROBABILITY T= 5 Computation D= 3 General ***Multiple Parts***

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Based upon item submitted by D. Kleinbaum - Univ of North Carolina Numerical Answer SIMPLEPROBABILIT CONDITIONALPROBA PROBABILITY T=10 Computation D= 3 General Biological Sciences ***Calculator Necessary***

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Based upon item submitted by D. Kleinbaum - Univ of North Carolina Numerical Answer SIMPLEPROBABILIT CONDITIONALPROBA PROBABILITY T=10 Computation D= 4 General ***Calculator Necessary***

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Based upon item submitted by A. Matthews - U. Mass. Numerical Answer CONDITIONALPROBA SIMPLEPROBABILIT JOINTPROBABILITY PROBABILITY T= 5 Computation D= 3 General Biological Sciences ***Multiple Parts***

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Item is still being reviewed Short Answer EVENTS CONDITIONALPROBA PROBABILITY T= 5 Comprehension D= 5 General Biological Sciences

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Based upon item submitted by H. B. Christensen - BYU Multiple Choice NONBAYESIAN CONDITIONALPROBA PROBABILITY T= 5 Computation D= 1 General Business

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Based upon item submitted by F. J. Samaniego - UC Davis Multiple Choice NONBAYESIAN CONDITIONALPROBA PROBABILITY T= 5 Computation D= 2 General

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Based upon item submitted by F. J. Samaniego - UC Davis Multiple Choice SIMPLEPROBABILIT NONBAYESIAN PROBABILITY CONDITIONALPROBA T= 2 Computation D= 2 General

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Based upon item submitted by D. Kleinbaum - Univ of North Carolina Multiple Choice NONBAYESIAN CONDITIONALPROBA PROBABILITY T= 2 Comprehension D= 3 Biological Sciences

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Item is still being reviewed Numerical Answer NONBAYESIAN CONDITIONALPROBA PROBABILITY T= 2 Application D= 3 General ***Multiple Parts***

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Based upon item submitted by W. Geeslin - UNH Numerical Answer NONBAYESIAN SIMPLEPROBABILIT CONDITIONALPROBA PROBABILITY T= 5 Computation Application D= 3 General ***Multiple Parts***

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Based upon item submitted by D. Kleinbaum - Univ of North Carolina Numerical Answer SIMPLEPROBABILIT NONBAYESIAN PROBABILITY CONDITIONALPROBA T=10 Computation D= 2 General Biological Sciences ***Multiple Parts***

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Item is still being reviewed Multiple Choice STANDARDDEVIATIO DESCRSTAT/P PARAMETRIC STATISTICS T= 2 Comprehension D= 2 General

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