25-2 Which of the following pairs of events are mutually exclusive?a. b. c. d. e. A: A: A: A: A: the odd numbers; the even numbers; the numbers less than 5; the numbers above 100; negative numbers; B: B: B: B: B: the number 5 the numbers greater than 10 all negative numbers the numbers less than -200 odd numbers d. A: the numbers above 100; B: the numbers less than -200 27-1 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) (b) (c) (d) independent mutually exclusive random variables randomly independent. (b) mutually exclusive. 34-1 Suppose a certain ophthalmic 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 relationship between the events A = a person has brown eyes and B = a person has blue eyes? (circle the correct answer) i. independent iii. simple A. 90/300 = .3 ii. exhaustive iv. mutually exclusive Chance Process Definition: A process producing results or outcomes where the next outcome cannot be specified in advance.25 0 <= Probability <= 1. 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. correct it. iv. 40-1 Define the following term and give an example of its use.B.12 C. A: True. The law of large numbers makes it possible to predict long run relative frequencies but not particular chance events. Your example should not be one given in class or in a handout. 44-2 Which of the following is NOT a possible probability? a. d.0 71-1 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: . mutually exhaustive 37-2 True or False? If False. 1. Example: If we were to regard dropping a baited line into water suspected of harboring flounder as a random process. b. 25/100 1. but where the long term chances of that outcome can be determined. (120/300)*(90/300) = . 38-2 Define the complement to the event A. c. but that if we repeatedly drop a properly baited line it will result in a catch some percentage of the time. The complement of A is the set of elements which do not belong to A.25 1 0 b. a. 300/1000 (90 + 35 + 121 + 54)/1000 = 300/1000 533-2 Among twenty-five articles.$99 | $100-$499 | $500 or more --------------------------------------------------------------- .08 a.DISEASE STATUS Tall HEIGHT Medium Short None Mild Moderate Severe Totals |-------|---------|------------|----------| | 122 | 78 | 139 | 61 | 400 |-------|---------|------------|----------| | 74 | 51 | 90 | 35 | 250 |-------|---------|------------|----------| | 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 b. 300/600 e. 1/3 P[MD/D] = (3/25)/(9/25) = 3/9 = 1/3 534-1 The checking accounts of Save-More Bank are categorized by age of account and balance in account. Balance | | | Age of Account | $0 . 1/3 . nine are defective. six having only minor defects and three having major defects.25 . d. Determine the probability that an article selected at random has major defects given that it has defects. 300/1000 d. We are going to select an account at random from this group of 2000 accounts.24 . c. 300/500 c. b. 800/1000 c. 3 years$0 .200/2000 = 3/4 iii) c) 11/20 P($100 or more) = P($100 .P(($0 . State G'.3 years) = 700/1200 = 7/12 v) The two variables are not independent because P(0 .$99) or (3 years or more)) = P($0 . 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.$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.$99) =/= P(0 .$499) + P($500 or more) = 500/2000 + 600/2000 = 11/20 iv) d) 7/12 P($0 .3 years$100 .$99) + P(3 years or more) .$990 . i) c) 1/3 P($500 or more0 . the complement of G.less than 3 years | 700 | 100 | 400 3 or more years | 200 | 400 | 200 --------------------------------------------------------------i) Then P($500 or more0 . Let G be the event: "At least five accounts are less than 3 years old".3 years) = 400/1200 = 1/3 ii) c) 3/4 P(($0 .3 years$500 or more) .$99) and (3 years)) = 900/2000 + 800/2000 .$499) =/= P(0 .3 years) = a) 6/7 b) 2/3 c) 1/3 d) 1/5 e) none of these ii) Then P[($0 . It is third down.(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) 538-1 The Green Bay Packers and Chicago Bears are playing their annual charity game. four yards to go for the Packers on their own 45 yard line.P(Male and 31 or more) = (1200/2000) + (600/2000) . with probabilities as indicated: . Based on previous performances by the Packers. the next play will see one of the following occurrences. 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 ii) Then P[Male or (31 or more)] = a) 1/5 b) 3/10 c) 1/2 d) 7/10 iii) Then P(Female) = a) 3/10 b) 2/5 c) 3/5 d) 2/3 e) none of these e) none of these e) none of these iv) What is the conditional probability that the depositor drawn is 30 or less. We are going to select an individual at random from this group of 2000 depositors.vi) G': "At least 10 accounts are three years old or older" or "less than five accounts are less than three years old" 536-1 The depositors at Save-More Bank are categorized by age and sex. 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? 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). 03 = .1375 = 10/400 + 10/400 + 20/400 + 15/400 . quarterback sacked) = .5000 = 145/400 + 55/400 . find for a randomly selected individual from this population the probability that he or she: a) b) c) d) e) f) g) h) Is in the age interval 40-49 Is in the age interval 40-49 and weighs 170-189 lbs Is in the age interval 40-49 or 60-69 Is in the age interval 40-49 or 60-69 and weighs 150-169 lbs Is in the age interval 40-49 given a weight between 150-169 lbs Weighs less than 170 lbs Weighs less than 170 lbs and is less than 50 years Weighs less than 170 lbs given that he is less than 50 years a) b) c) d) e) f) g) .35 b) .15/.P P P P P P P (pass incomplete) = .2250 = 30/400 + 60/400 .48 c) .03 (run for first down) = .20 = .1250 = 50/400 .05 (interception) = .3625 = 145/400 .15 + .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) a) .48 = .05 + .15 (pass complete.25 + .3125 539-1 400 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.25 (whistle is blown before play) = .20 (busted play.2500 = 15/60 .15 + .25 (pass complete for first down) = .0625 = 15/400 + 10/400 . no first down) = . What is the probability of exposure in the group? b.0417 . P(diseaseexposure present) = 75/400 = . Compute the probability of disease being present conditional on the presence of exposure and conditional on the absence of exposure.h) . The results are presented in the following table: | LEFT 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 RIGHT EYE For a randomly selected person from the population sampled above. P(exposure) = 400/1000 = .05 c) 35/3750 = .750 women aged 30-39 was measured for unaided distance vision in both right and left eyes.0093 541-1 The following two-way table shows the frequencies of occurrence of a hypothetical exposure and disease in a group of 1000 people.40 b. 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 a) 1245/3750 = .2340 = (10+10+20+15)/(145+90) 540-1 Each member of a sample of 3. What is the joint probability of both exposure and disease being present in the group? c. Exposure Present Absent Disease Present Absent | 75 325 | 400 25 575 | 600 --------------------------100 900 1000 a. a.3320 b) 20/400 = .075 c.1875 P(diseaseexposure absent) = 25/600 = . P(exposure and disease) = 75/1000 = . A student is selected at random. given that he has the disease.80.25 546-1 A dormitory on campus houses 200 students. Classified according to rank: Apprentice Journeyman Master ------------------------------------ .24 (b) . is .25 P(MDD) = P(MD and D)/P(D) = (2/25)/(8/25) = .542-1 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. therefore you cannot evaluate whether the two events are independent or not. Do you agree with him? Why or why not? You don't know whether P (living near tracks disease) is equal to P(living near tracks). 50 are upper division students. one can construct the following table: Lower Upper Male | 80 | 40 | 120 ------------------------------------Female | 70 | 10 | 80 ------------------------------------| 150 | 50 | 200 P(LowerFemale) = 70/80 549-1 A local trade union consists of plumbers and electricians.25 (c) 1/3 (d) . 120 are male. and 40 are upper division male students. 545-1 Among twenty-five articles eight are defective. given the student is a female.08 (b) . The probability of selecting a lower division student. Determine the probability that an article selected at random has major defects given that it has defects. is: (a) 7/8 (b) 7/15 (c) 2/5 (d) 7/20 (e) 1/4 (a) 7/8 Given the above information. six having only minor defects and two having major defects. (a) . c. Pr(P'|C) = . 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. Among the results.e. none of these Pr(PC) = .1 4 5 6 0. b.40 e. Pr(P|C') = . 553-2 Suppose a loaded die has the following model: Face Probability 1 2 3 0. Pr(C|P) = . P = attended party) a.40 d.1 0. 4/15 20/75 = 4/15 550-1 In an effort to get at the source of an outbreak of Legionnaire's disease at the 197? APHA convention. What is the probability that the die shows a four? b.1 0. c. P(4|odd number) = 0 .40 is the correct notation. Which of the following statements is appropriate for describing the 40% of cases who went to the party? (C = case. d. the probability that he is a journeyman is: a. Given that the person selected is a plumber. a team of medical detectives (i. none of these c. 1/2 1/3 4/15 2/15 none of these. What is the probability that the die shows a 1? a.Plumbers Electricians 25 | 20 | 30 -----------------------------------15 | 40 | 20 ------------------------------------40 60 50 75 75 A member of the union is selected at random. whereas 10% of the controls attended the same party.3 If this die is thrown and the top face shows an odd number. a. epidemiologists) carried out a casecontrol study involving all 50 cases and a sample of 200 non-cases out of the 4000 persons attending the convention.3 0. e. Pr(C|P') = .1 0.40 e.40 b. a marble is drawn from Urn H. A coin is to be tossed.1 + .1) = . P(heads and red) d. P(heads|red) (a) P(heads. If it lands tails a marble is drawn from Urn T. P(tails) c.3/(. P(blue) b. P(1|odd number) = P(1 and odd number)/P(odd number) = .b. Find the following probabilities: ------------------| H | T | ----------------------------R | 2/6 | 1/6 | 1/2 ----------------------------B | 1/6 | 2/6 | 1/2 ----------------------------| 1/2 | 1/2 | 1 ------------------------a.3 + . Urn T contains 1 red and 2 blue marbles. Urn H contains 2 red marbles and 1 blue marble. 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) = 2/3 560-1 A sample of 2000 individuals is distributed according to eye color and the presence or absence of a certain opthalmic trait as follows: TRAIT | | Blue EYE COLOR Brown Other | | . If it lands heads.6 558-1 There are two urns marked H and T. P(red) e. 350/1200 = 7/24 d.550/2000 = 1730/2000 = 173/200 . what is your estimate of the probability that: a. (600 + 480)/2000 = 27/50 f. the person has neither brown nor blue eyes given that the trait is absent? _______________ d. the person does not have the trait or does not have brown eyes? _______________ a. the person does not have brown eyes? _______________ f. the person has blue eyes? _______________ b.---------------------------------------------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. the person has blue eyes or has neither blue nor brown eyes? _____________ g. the person has neither brown nor blue eyes and the trait is present? _______________ e.0 = 1080/2000 = 27/50 g. P(Blue union Other) = P(Blue) + P(Other) .P(Blue and Other) = 600/2000 + 480/2000 . P(No trait union (Blue or Other)) = 1200/2000 + 1080/2000 . 600/2000 = 3/10 b. the trait is present and the person has brown eyes? ____________ c. 130/2000 = 13/200 e. 270/2000 = 27/200 c.