Curtin University Mathematics Final Exam Exercise PDF

Tutorial 6.0 Page 1 Tutorial 6.0 Page 2 Tutorial 6.1 Page 1 Tutorial 6.1 Page 2  Extra Exercise 1. Given A = [ ],B=[ ],C=[ ] and D = [ ], determine each of the following. If any cannot be determined, st...

Tutorial 6.0 Page 1 Tutorial 6.0 Page 2 Tutorial 6.1 Page 1 Tutorial 6.1 Page 2  Extra Exercise 1. Given A = [ ],B=[ ],C=[ ] and D = [ ], determine each of the following. If any cannot be determined, state clearly. (a) A + B (b) A + C (c) C – A (d) 2D (e) 3B (f) B + D (g) 2A (h) 2A – C 2. A is a 3 × 2 matrix, B is a 3 × 2 matrix, C is a 2 × 3 matrix and D is a 1 × 3 matrix. State the dimensions of each of the following products. For any that cannot be formed, state this clearly. (a) AB (b) BA (c) BC (d) CB (e) AD (f) DA (g) BCA (h) DAC 3. Calculate the determinant of [ ] using expansion by cofactors method. 4. Determine the inverse of [ ] using Gauss-Jordan Method.  Tutorial 7.0 Topic: Vector Operations and Unit Vector 1. Determine the magnitude of the following vectors. 𝑖 𝑗 (a) ⃗⃗⃗⃗⃗ 𝑃𝑄 = 5 − 7 ~ ~ 𝑣 2 (b) = ( ) ~ −4 5 ⃗⃗⃗⃗⃗ = (−7) (c) 𝑅𝑆 3 2. Determine the direction of the vectors below. Give your directions to the nearest degrees. 𝑟 𝑖 𝑗 (a) =5 − 7 ~ ~ ~ 𝑠 𝑖 𝑗 (b) =-7 + 2 ~ ~ ~ 𝑣 −2 (c) = ( ) ~ −5 𝑖 𝑗 𝑘 3. If |2 + 𝛽 − 4 | = 6, calculate the possible values of the constant 𝛽. ~ ~ ~ 𝑖 𝑗 𝑘 ⃗⃗⃗⃗⃗ 𝑖 𝑗 𝑘 𝑖 𝑗 𝑘 4. Given that ⃗⃗⃗⃗⃗ 𝐴𝐵 = 2 −4 +5 , 𝐵𝐶 = 3 +6 −2 , ⃗⃗⃗⃗⃗ 𝑃𝑄 = 2 +3 −6 , and ~ ~ ~ ~ ~ ~ ~ ~ ~ 𝑖 𝑘 ⃗⃗⃗⃗⃗ = 5 −7. Determine 𝑄𝑅 ~ ~ (a) 𝐴𝐶⃗⃗⃗⃗⃗ (b) ⃗⃗⃗⃗⃗ 𝑃𝑅 𝑎 ⃗⃗⃗⃗⃗ = 𝑐. S is the on point on AB such that 5. OABC is a parallelogram with ⃗⃗⃗⃗⃗𝑂𝐴 = and 𝑂𝐶 ~ ~ AS : SB = 3:1, and T is the point on BC such that BT : TC = 1:3. 𝑎 𝑐 𝑎 𝑐 (a) Express each of the following in terms of or or and ~ ~ ~ ~ (i) ⃗⃗⃗⃗⃗ 𝐴𝐶 (ii) ⃗⃗⃗⃗⃗ 𝑆𝐵 ⃗⃗⃗⃗⃗ (iii) 𝐵𝑇 (iv) ⃗⃗⃗⃗ 𝑆𝑇 (b) Explain why ⃗⃗⃗⃗ 𝑆𝑇 and ⃗⃗⃗⃗⃗ 𝐴𝐶 are parallel, and state the value of the ratio ST : AC. Page 1  Tutorial 7.0 𝑖 𝑗 6. The centroid of the triangle OAB is denoted by G. if O is the origin and ⃗⃗⃗⃗⃗ 𝑂𝐴 =4 +3 , ~ ~ 𝑖 𝑗 ⃗⃗⃗⃗⃗ =6 −. Write 𝑂𝐺 𝑖 𝑗 ⃗⃗⃗⃗⃗ in terms of the unit vectors, and. 𝑂𝐵 ~ ~ ~ ~ (Hint: The centroid of the triangle divides the medians in the ratio 2 : 1. A median joins a vertex with the midpoint of the opposite side.) 3 7. The position vectors of the point A, B and C are represented as column vectors as ( ) , 7 9 7 ⃗⃗⃗⃗⃗ = 𝑚𝑂𝐴 ⃗⃗⃗⃗⃗ + 𝑛𝑂𝐵 ⃗⃗⃗⃗⃗. ( ) and ( ) respectively. Determine the values of m and n if 𝑂𝐶 −3 9 8. A is the point (3, 2) and B is the point (1, 5). If P is a point on AB such that AP : PB = 1:2, determine the position vector of P. 5 7 11 9. Point 𝑃, 𝑄 and 𝑅 have position vectors (4) , (5) and ( 7 ) respectively. 1 4 10 ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ (a) Evaluate 𝑃𝑄 and 𝑄𝑅. (b) Deduce that 𝑃, 𝑄 and 𝑅 are collinear and calculate the ratio 𝑃𝑄 ∶ 𝑄𝑅. 10. Given that 𝑃(2, 13, −5), 𝑄(3, 𝛽, −3) and 𝐶(6, −7, 𝛾) are collinear, determine the values of the constants, 𝛽 and 𝛾. 11. Evaluate the unit vector in the direction of the following vectors. 𝑞 𝑖 𝑗 (a) =8 − 6. ~ ~ ~ −3 (b) m = ( 12 ) ~ −4 𝑎 𝑖 𝑗 12. Calculate the magnitude of the vector ~= 5 +12 and hence obtain the vector, which is ~ ~ 𝑎 in the same direction as ~ but having a magnitude of (a) 2 units. (b) 5 units. After that, state the vector, which has a magnitude of 5 units but in the opposite direction 𝑎 of vector ~. 𝑖 𝑗 13. Evaluate a vector of magnitude √5 in the direction of a vector v = 4 − 8. ~ ~ ~ Page 2  Tutorial 7.1 Topics: Scalar Product of Vectors ; Parallel and Perpendicular vectors ; Vector Equations of straight Lines 1. Determine the scalar product of the following vectors. 4 2 (a) (−2) and ( 1 ) 1 −3 ⃗⃗⃗⃗⃗ = (−4) and ⃗⃗⃗⃗⃗ (b)𝑃𝑄 −4 𝑅𝑆 = ( ) −5 3 2 −3 (c) ~𝑝 = (−1) and ~𝑞 = ( 1 ) 4 5 2. If ~𝑝 = −3 ~𝑖 + 4 ~𝑗 , ~𝑞 = −2 ~𝑖 − 5 ~𝑗 , 𝑟 ~ = 5 ~𝑖 + 2 ~𝑗. Determine the values of the following dot products. (a) ~𝑝. ~𝑞 (b) ~𝑞. ~𝑟 (c) (~𝑝 + ~𝑞 ). ~𝑟 (d) (~𝑝 + ~𝑞 ). (~𝑟 − ~𝑞 ) 3. Evaluate the angle between the given vectors in each of the following. Give all your answers in degrees accurate to one decimal place. (a) 3 ~𝑖 − 5 ~𝑗 − 2 ~𝑘 and ~𝑖 − 6 ~𝑘 −2 4 (b) p = ( 1 ) and q = (−3) ~ 3 ~ 3 2 3 4. Calculate the cosine of the angle between the vectors u = ( 3 ) and v = (−5). Give your answer in surd ~ ~ −1 2 form. 5. Given that the point A is (2, 3), B is (3, -10) and C is (k, 2), determine the possible value of k if angle ACB is a right angle. 𝑡 𝑡+2 6. Given two vectors, u = ( 4 ) and v = (1 − 𝑡) are perpendicular to each other. Determine the possible ~ ~ 2𝑡 + 1 −1 values of the constant t. 2 7. If p = ( 6 ) , q = ( 𝜆 ) and r = (𝜇). Calculate the value of the ~ −1 ~ −8 ~ (a) constant 𝜆, given that p and q are parallel in the same direction. ~ ~ (b) constant 𝜇, given that p and r are perpendicular to each other. ~ ~ Page 1  Tutorial 7.1 8. Identify whether the given two vectors are parallel in the same direction or perpendicular to each other or neither both. Justify your answers by determining the value of their scalar products. 2 4 (a) a = ( ) and b = ( ) ~ 8 ~ −1 −3 6 (b) s `= ( ) and t = ( ) ~ 1 ~ −2 6 9 (c) ⃗⃗⃗⃗⃗ 𝐴𝐵 = (−8) and ⃗⃗⃗⃗⃗ 𝐵𝐶 = (−12) 2 3 5 3 (d) r = (−6) and s = (−2) ~ ~ 2 1 9. A point P(3, 2) on a Cartesian plane moves in the direction (4 ~𝑖 − 2 ~𝑗 ). Evaluate the position vector of any point on the path taken by P. 10. Contruct the vector equation of a line as a column vector which (a) passes through (3, -2) and parallel to − ~𝑖 + 3 ~𝑗. (b) passes through (-1, 5) and parallel to 2 ~𝑖 + 6 ~𝑗. 11. The position of a ship S at 10.00 am is (2 ~𝑖 + 5 ~𝑗 ) km. If it moves with a velocity of (−3 ~𝑖 − 4 ~𝑗 ) km/h, calculate its position vector for (a) 3 hours later. (b) 𝑡 hours later. 12. Identify whether the points P(3, 2) is lying on the line with vector equation, 𝑟 ~ = (2 ~𝑖 + 4 ~𝑗 ) + 𝑡(3 ~𝑖 − 2 ~𝑗 ) where t is a scalar. 3 + 4𝑡 13. Given a vector equation of a line written as column vector, ~𝑟 = ( ) where t is a scalar. Write down the 2 − 2𝑡 vector equation in parametric form and then convert it into Cartesian equation of a line. Page 2 Extra Exercise - Vector Equation of a Line Extra Exercise - Vector Equation of a Line Copyright of Curtin University Tutorial 8.0 Topic: Counting Technique 1. If the Eagles, Collingwood and Hawthorn finish as the top three teams in the AFL, how many possible finishing orders are there for these three teams? 2. Suppose that the undergraduate students in three departments – geography, history and psychology – are to be classified according to gender and year (1- 4) in school. How many categories are there? 3. In how many ways can three officers – president, secretary and treasurer – be selected in order from a club that has 20 members? 4. In how many ways can a sum greater than five be obtained when tossing a pair of dice? 5. A social committee selects 20 records to be played at a social. In how many orders can these 20 records be played? 6. When buying a new car there is a choice of Automatic or Manual; Sedan, Wagon or Hatchback; and five colors. How many different ‘versions’ of the car are there? 7. A multiple-choice test of forty questions has five alternatives for each question. How many different ways are there of answering the paper? 8. In how many ways can the five letters A, B, C, D and E can be arranged in a row? 9. Shown below is a café blackboard menu. Soup: Chilled Soup or Vegetable Soup Main Course: Grilled Fish, Mixed Grill or Fillet Steak Dessert: Black Forest Cake or Pavlova How many possible combinations can be chosen if a meal consists of one choice from each course? Page 1 Copyright of Curtin University Tutorial 8.0 10. How many different arrangements can be formed from the words below? (a) MISSISSIPPI (b) COMPUTER (c) PROVIDE (d) PROGRAMMERS (e) APPLE 11. There are 8 candidates for Head Girl and 5 for Head Boy. (a) How many orders are possible for the names on the ballot paper for Head Girl? (b) How many orders are possible for the names on the ballot paper for Head Boy? (c) How many different pairs of Head Boy and Head Girl are there? 12. An exam paper consists of parts A and B. In part A you must complete three questions out of the five and in part B you must complete two questions out of four. How many different ways are there of completing the paper? 13. At graduation night awards are to be given to the top three students in year 12. If there are 180 students in year 12 how many ways can the awards be given if: (a) all three awards are the same, (b) there are different awards for 1st, 2nd and 3rd? 14. A social committee selects 30 records to be played at a social from a DJ’s collection of 300. (a) In how many orders can these 30 records be played? (b) In how many ways can this set of 30 records be chosen? 15. In horse racing, three common types of bets are: A: Quinella – selecting the 1st and 2nd horse in any order. B: Tierce – selecting the 1st, 2nd and 3rd horse in any order. C: Trifecta – selecting the first three horses in the order they finish. In a race with 14 horses how many different: (a) Quinellas (b) Tierces (c) Trifectas are possible, if dead heats do not occur? Page 2 Copyright of Curtin University Tutorial 8.0 16. The positions of chairman, vice chairman, secretary and treasurer are to be filled from 35 available candidates. How many ways can this be done if no candidate can fill more than one of the positions? 17. How many permutations are there of the letters of the word OUTWEIGH chosen three at a time? 18. From a committee of 12 people a subgroup of 5 is to be formed to represent the committee at a particular function. How many different such subgroups are possible? 19. How many committees consisting of four women and four men can be chosen from a group of seven women and eight men? 20. The letter of the word LONDON is arranged in a straight line. In how many ways the arrangements can be formed if: (a) it starts and end with the letter O. (b) the consonant all together. 21. A team of 3 students is being chosen from a committee of 15. The committee consists of 10 boys in which two of them are twin and 5 girls. Determine the number of different team can be formed if: (a) the twin must be in the team. (b) the team must contain at least one boy and at least one girl. (c) only one of the twin must be in the team. Page 3 Extra Exercise 1. A bank allows its customers to choose their ATM pin number either with 4 digits or 4 letters. How many different choices are available to customers? 2. Seven files, A, B, C, D, E, F and G are to be arranged on a shelf. In how many ways can this be done? In how many ways of these arrangements is file A next to file B? 3. In how many ways can Abel, Joey, Chai, Samantha, Penny and Russell be seated in a row of six seats if Abel and Russell want to sit side by side? 4. If no number contains repeated digits, how many odd numbers greater than 40,000 can be formed by choosing from the digit 1, 2, 3, 4 and 5? 5. In our metropolitan phone system, a number consists of seven digits. Given that the first digit cannot be a zero, how many different phone numbers exist for our system? Page 1 of 2 Answers 1. 466976 2. 5040; 1440 3. 240 4. 30 5. 9,000,000 Page 2 of 2 Copyright of Curtin University Tutorial 9.0 Topic: Probability I 1. Evaluate the probability of choosing an even number from the set of numbers {2, 3, 4, 5, 6, 7, 8, 9}. 2. An unbiased octahedral die has the numbers 1 to 8 on its eight faces, one number to a face. This die is thrown. Determine the probability of obtaining (a) an even number. (b) an odd number. (c) a prime number. 3. Each arrangement of the seven letters of the word OSMOSIS is put on a slip of paper and placed in a hat. One slip is drawn at random from the hat. Determine the probability that the slip contains an arrangement of the letters with an O at the beginning and an O at the end. 4. If the probability that choosing odd numbers greater than 3 from a set of odd numbers is 0.4, determine the probability of choosing odd numbers at most 3. 5. Given E is selecting a month and getting a month that begins with J. List all the complement of the event E and then calculate its probability. 6. A fair cubical die is rolled. Count the probability that the number obtained is (a) a multiple of 3. (b) a factor of 6. (c) less than 5. (d) at least 5. 7. If a letter is chosen at random from the English alphabet, calculate the probability that the letter is (a) a vowel. (b) listed somewhere ahead of letter J. (c) listed somewhere after of the letter G. Page 1 Copyright of Curtin University Tutorial 9.0 8. A store has 6 TV Graphic magazines and 8 Newstime magazines on the counter. If two customers purchase a magazine each, calculate the probability that one of each magazine was purchased. 9. The five letters of the word ‘TIMES’ are written on five cards, with one letter on each card. The five cards then shuffled and two of the cards are dealt face up in a line to form a ‘word’. Determine the probability that word so formed (a) is the word ‘IT’ (b) contains only vowels. (c) contains only consonants. 10. An octahedral die has the numbers 1 to 8 on its eight faces, one number to a face. This die is rolled and a coin is tossed simultaneously. Determine the probability of obtaining a ‘7’ on the die or a tail on the coin. 11. Toss three coins and determine the probability of getting at least one head. 12. A jar contains 240 sweets of orange, lychee and coffee flavour. There are 90 orange flavoured sweets. If a sweet is picked at random from the jar, the probability of picking a 1 lychee flavoured sweet is. How many coffee flavoured sweets are there? 3 13. What is the probability of getting an odd number or a prime number with one toss of a die? 14. Given that 𝑃(𝐴) = 0.23 and 𝑃(𝐵) = 0.45, evaluate 𝑃(𝐴 ∪ 𝐵)′ if A and B are mutually exclusive events. 15. An interviewer is assessing the performance of candidates. His database of 50 candidates shows that 14 are fluent in English (E), 15 are fluent in Mandarin (M) and none are fluent in both languages of E and M. If a new interviewer takes over and does not consult the database records thoroughly, calculate the probability that he will choose a candidate who will be fluent in English or Mandarin. Draw a relevant Venn Diagram depicting the number of each category and then calculate the required probability using the Addition Law of Probability. Page 2 Copyright of Curtin University Tutorial 9.1 Topic: Probability II 1. An urn contains two blue marbles (say B1 and B2) and two white marbles (W1 and W2). If two marbles are randomly drawn without replacement, calculate the probability that the second marble drawn is white, given that the first marble drawn is blue. [2/3] 2. In a group of 50 students, 20 take an English subject, 25 take a technology subject and 15 take both. Determine the probability of randomly selecting a student who is taking: (a) the technology subject, given that the student is already taking English. [3/4] (b) the English subject, given that the student is already taking technology. [3/5] 3. A number is picked at random from the digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Given that the number is a multiple of 3, determine the probability that the number is (a) even. [1/2] (b) a multiple of 4. [1/4] 4. The probability that it rains and the bus arrives late is 0.023, John hears the weather forecast, and there is a 40% chance of rain tomorrow. Calculate the probability that the bus will be late, given that it rains. [0.0575] 5. If P(E) = 1 3 , P( E  F ) = 15 8 , and P( E  F ) = 15 , calculate each of the following: (a) P(F | E). [3/5] (b) P(F). [2/5] (c) P(E | F). [1/2] (d) P(E | F’). [2/9] 6. A number of students were grouped according to their reading ability and education. The table below shows the results. Education Reading ability Low Average High Graduated high school 6 18 43 Did not graduate 27 16 7 If a student is selected at random, determine these probabilities. (a) The student has a low reading ability, given that the student is a high school graduate. [6/67] (b) The student has a high reading ability, given that the student did not graduate. [7/50] 7. Because of gypsy month infestation of three large areas that are densely populated with trees, consideration is being given to aerial spraying to destroy larvae. A survey was made of the 200 residents of these areas to determine whether or not they favor the spraying. The resulting data are shown in the table below. Suppose that a resident is randomly selected. Let I be the event “the resident is from Area I” and so on. Determine each of the followings: (a) P(F) (b) P(F | II) (c) P(O | I) [5/8; 35/58; 11/39] (d) P(III) (e) P(III | O) (f) P(II | N’) [8/25; 10/47; 25/86] Area I Area II Area III Total Favor (F) 46 35 44 125 Opposed (O) 22 15 10 47 No opinion (N) 10 8 10 28 Total 78 58 64 200 Page 1 Copyright of Curtin University Tutorial 9.1 8. At a large factory, the employees were surveyed and classified according to their level of education and whether or not they smoked. The data are shown in the table below. Educational level Smoking habit Not high school High school College graduate graduate graduate Smoke 6 14 19 Do not smoke 18 7 25 If an employee is selected at random, calculate these probabilities. (a) The employee smokes, given that he or she graduated from college. [19/44] (b) Given that the employee did not graduate from high school, he or she is a smoker. [1/4] 9. A fair cubical die is thrown twice. Determine the probability of obtaining a ‘4’ on the first throw and an odd number on the second throw. [1/12] 10. A die is thrown five times, calculate the probability of getting number less than 3 on all the five throws. [1/243] 11. The Reader’s Digest global poll found that more than half (51%) of married American women wish their husbands were thinner. If three married American women are selected at random, determine the probability that all three will say that they wish their husbands were thinner. (Source: Reader’s Digest February 2010) [0.133] 12. Approximately 9% of men have a type of colour blindness that prevents them from distinguishing between red and green. If four men are selected at random, determine the probability that all of them will have this type of red-green colour blindness. [0.0000656] 13. Suppose the probability of the event “Bob lives 20 more years” (B) is 0.8 and the probability of the event “Doris lives 20 more years” (D) is 0.85. Assume that B and D are independent events. Determine the probability that (a) Bob and Doris live 20 more years. [0.68] (b) at least one of them lives 20 more years. [0.97] (c) exactly one of them lives 20 more years. [0.29] 14. If events E and F are independent with P(E) = 2/5 and P(E ∩ F) = 1/3, evaluate P(F). [5/6] 15. An urn contains six chips numbered from 1 to 6. A chip is randomly drawn. Let E be the event of drawing a 4 and F be the event of drawing a 6. (a) Are E and F mutually exclusive? Justify your answer. (b) Are E and F independent? Justify your answer. 16. The probability that a student is late for school on any day is 0.05. Determine the probability that the student is (a) late for school on exactly two consecutive schooling days. [0.008574] (b) late for school on any one of the five schooling days. [0.2036] 17. A bag contains ten marbles: 5 red, 3 blue and 2 green. Two marbles are randomly selected from the bag, one after the other, the first marble being replaced before the second is selected. Determine the following probabilities (a) P(red marble and blue marble in that order). [3/20] (b) P(red marble and blue marble in any order). [3/10] (c) P(two marbles of the same colour). [19/50] (d) P(two marbles of the different colour). [31/50] (e) P(second marble is red | first marble is blue). [1/2] Page 2 Extra Exercise 1. A committee of three people is randomly selected from Alice, Ben, Chad, Dee and Eric. What is the probability that Alice is on the committee? 2. In a large group of people, it is known that 10% have a hot breakfast, 20% have a hot lu n ch an d 25% have a hot breakfast or a hot lunch. Determine the probability that a person chosen at random from this group has hot breakfast and a hot lunch. 3. A parent – teacher committee consisting of four people is to be formed from 20 parents an d fiv e teachers. If the selection will be random, determine the probability that the committee will consist of (a) all teachers. (b) all parents. 4. A box contains 3 yellow golf balls, 2 blue golf balls and 6 white golf balls. Evaluate the probability of selecting 5 golf balls at a time in which at least 3 of them are white. 5. Event A is that of rolling a die and getting an even number. Event B is that of rolling a die and getting a number less than 5. Prove that A and B are independent events. 6. If events E and F are independent with P(E) = 1/3 and P(F) = ¾, evaluate each of the followings: (a) P(E ∩ F) (b) P(E U F) (c) P(E | F) (d) P(E’| F) (e) P(E ∩ F’) (f) P(E U F’) (g) P(E | F’) 7. Suppose we have a jar that contains five white, seven green, and nine red marbles. If one marble is drawn at random from the jar, determine the probability that it is white or green. 8. A fair cubical die is thrown. If event A is getting number less than three an d event B is getting number at least three, calculate the probability of event A and event B. 9. A company employs 93 people of whom 38 are male. 22 of the employees drive to wo rk an d 15 of these 22 are female. If one of the 93 employees is chosen at random, determine the probabilit y that they are (a) female. (b) a male who drives to work. (c) male given that they drive to work. (d) someone who drives to work given that they are male. 10. After the initial production run of new style of steel desk, a quality control technician foun d t h at 40% of the desks had an alignment problem and 10% had both a defective paint job and an alignment problem. If a desk is randomly selected from this run, and it has an alignment problem, what is the probability that it also has a defective paint job? 11. Events A and B are independent events with P(A) = 0.2 and P(B) = 0.25. Evaluate the following probabilities. (a) P(A | B) (b) P(B | A) (c) P(A  B) (d) P(A  B)’ 12. Forty percent of the students at a particular university are male. Eighty percent of the males and 40% of the females are studying commerce course. If one student is randomly selected from the students at this university, calculate the probability that the student is (a) female. (b) studying commerce course. (c) male given that they are studying commerce course. (d) female given that they are not studying commerce course. Page 1 of 2 Answers 1. 0.6 2. 0.05 3. (a) 1/2530 (b) 969/2530 4. 281/462 5. Yes, they are independent to each other. 6. (a) ¼ (b) 5/6 (c) 1/3 (d) 2/3 (e) 1/12 (f) ½ (g) 1/3 7. 4/7 8. 0 9. (a) 55/93 (b) 7/93 (c) 7/22 (d) 7/38 10. 0.25 11. (a) 0.2 (b) 0.25 (c) 0.05 (d) 0.6 12. (a) 0.6 (b) 0.56 (c) 4/7 (d) 9/11 Page 2 of 2 Copyright of Curtin University Tutorial 10.0 Topic: Probability Distribution I 1. Determine which of the following are measured in discrete quantities: (a) total time taken to run 200m. (b) total goals in a football match. (c) total absentee in a class. (d) total distance between two towns. (e) the weight of student. 2. Identify whether the following are probability distribution tables (p.d.f tables). (a) 𝑥 1 2 3 4 𝑃(𝑋 = 𝑥) 0.2 0.4 0.1 0.3 (b) 𝑡 0 1 2 3 𝑃(𝑇 = 𝑡) 0.5 -0.4 0.8 0.1 3. Identify whether the following are p.d.f. (a) P(X = x) = 0.1 for X = {1,2,3,4,5,6,7,8,9,10} (b) P(X = x) = 0.2x for X = {2,4,6} 4. Which of the following functions represent discrete probability functions? (a) f(x) = 0.0025x2 for x = 0,1,2,3,4,5,6,7,8,9,10 (b) f(x) = (0.025)3x for x = 0, 1, 2, 3 5. 𝑤 -3 -2 -1 0 1 𝑃(𝑊 = 𝑤) 0.1 0.25 0.1 0.3 s Consider the above probability distribution table, evaluate: (a) the value of s. (b) 𝑃(𝑊 ≤ 0). (c) 𝑃(−1 < 𝑊 ≤ 0). 6. The probability distribution for the random variable X as shown below. Page 1 Copyright of Curtin University Tutorial 10.0 𝑥 0 1 2 3 4 5 𝑃(𝑋 = 𝑥) 0.1 0.2 0.3 0.2 0.1 0.1 Determine: (a) 𝑃(1 < 𝑋 < 4) (b) 𝑃(𝑋 = 3|𝑋 > 2) (c) 𝑃(𝑋 = 5|𝑋 ≥ 3) (d) mode 7. A bag contains 5 marbles; 3 red and 2 blue. Suppose that three marbles are randomly selected from the bag, one after the other, each selected marble is not being returned to the bag after selection. Create a probability distribution table for the discrete random variable X, the number of reds this random selection process produces. 8. The discrete random variable has a p.d.f. 𝑐𝑥, for 𝑥 = 1,2,3,4 𝑃(𝑋 = 𝑥) { 0, for 𝑎𝑙𝑙 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥 Calculate the value of the constants c. Then, construct a probability distribution table for the above function. 9. Use tables of cumulative probabilities for the binomial distribution to determine these values: (a) P(X ≤ 4) where X ~ B(8, 0.3). (b) P(X ≤ 7) where X ~ B(20, 0.25). (c) P(X > 3) where X ~ B(9, 0.6). (d) P(X ≥ 4) where X ~ B(20, 0.3). 10. Given that X ~ B(4, 0.4). Evaluate the following probabilities: (a) P(X = 1). (b) P(X < 3). 11. Given that X ~ Poi(λ) per unit space and P(X = 0) = 0.1353. Evaluate: (a) λ (b) P(X = 1) (c) P(X = 2) (d) P(X < 4) (e) P(X > 2) (f) P(X ≥ 1) (g) P(X ≥ 3) (h) P(0 < X < 3 | X ≤ 2) (i) P(X < 2 | 0 < X ≤ 3) Page 2 Extra Exercise 1. The probability that a student selected at random from a class will pass an examination is 0.8. If 10 of the students in that class are selected at random, evaluate the probability that: (a) all of them will pass the examination. (b) more than 3 students will pass the examination. (c) none of the student will pass the examination. 2. The reception office at Tom’s Building Corporation receives an average of 4.7 p h o n e calls p er half an hour. Determine the probability that the number of phone calls receive at this time o ffice during a certain half an hour will be: (a) exactly 6. (b) less than 8. (c) 5 to 8. 3. An average of 3.2 employees of a telephone company are absent per day. Determine the probability that on a given day the number of employees who will be absent at this company is: (a) at least 7. (b) at most 3. 4. Each question of the multiple-choice test paper offers four answers, one of which is correct. A student answers 20 questions by randomly guessing which response is correct each time. If we define the random variable X as the number of the questions the student gets correct, determine the following probabilities accurate to 3 decimal places: (a) P (X = 5). (b) P (8 < X ≤ 10). 5. For a student to pass a particular course they must pass at least six of the eight tests. For each test the probability of the student passing is 0.9. Determine the probability of the student: (a) passes exactly 6 of the 8 tests. (b) pass the course. Answers 1. (a) 0.1074 (b) 0.9991 (c) 0.0000 2. (a) 0.1362 (b) 0.8960 (c) 0.4551 3. (a) 0.0446 (b) 0.6025 4. (a) 0.2023 (b) 0.0370 5. (a) 0.1488 (b) 0.9619 Page 1 of 1 Copyright of Curtin University Tutorial 11.0 Topic: Continuous Probability Density Functions 1. Determine which of the following are measured in continuous scale. (a) the number of kilograms of food a person consumes. (b) the lifetime of a wristwatch. (c) the systolic blood pressure for a certain group of obese people. (d) the average temperature of Miri. (e) the number of building permits issued each month in a certain city. 2. Identify whether each of the following is continuous p.d.f. (a) 𝑓(𝑥) = 0.2 for 2 < 𝑥 < 7 (b) 𝑓(𝑥) = 0.15𝑥 for 0 ≤ 𝑥 ≤ 4 3. Consider the graph of continuous probability density function above. Evaluate P(0.5 < 𝑋 < 1). Page 1 Copyright of Curtin University Tutorial 11.0 4. Referring the graph of continuous probability density function above, determine: (a) P(0 < 𝑋 < 4) (b) P(𝑋 ≥ 0) (c) P(2 ≤ 𝑋 < 5) 5. Given the probability distribution for the continuous random variable X is 𝑓(𝑥) = 0.02𝑥 for 0 ≤ 𝑥 ≤ 10. Determine: (a) P(1 < 𝑋 < 4) (b) P(𝑋 ≤ 3|𝑋 > 2) (c) P(𝑋 ≥ 5|3 ≤ 𝑋 ≤ 8) 6. Lisa is doing a biological survey of certain species of mosquito and finds that the probability density function for the length of its wing in mm is given by 𝑓(𝑥) = 0.05𝑥 + 0.15. Sketch the graph of the p.d.f and then use it to calculate: (a) the probability that the wing of a mosquito is less than 2mm long. (b) the probability of the wing of the mosquito is between 1.5mm and 3mm in length. Page 2 Copyright of Curtin University Tutorial 11.1 Topic: Normal Distribution 1. Given 𝑍~𝑁(0,1) , calculate: (a) 𝑃(𝑍 < 0.87) (b) 𝑃(𝑍 > −0.87) (c) 𝑃(1 < 𝑍 < 2) (d) 𝑃(−2 < 𝑍 < 2) (e) 𝑃(−0.493 < 𝑍 < 0.246) (f) 𝑃(|𝑍| < 2) (g) 𝑃(|𝑍| ≥ 0.87) 2. Given 𝑍~𝑁(0,1), determine the value of a if (a) 𝑃(𝑍 < 𝑎) = 0.507 (b) 𝑃(𝑍 < 𝑎) = 0.007 (c) 𝑃(𝑍 > 𝑎) = 0.0242 (d) 𝑃(𝑍 > 𝑎) = 0.812 (e) 𝑃(|𝑍| < 𝑎) = 0.837 (f) 𝑃(|𝑍| > 𝑎) = 0.24 3. Given 𝑋~𝑁(10,9). Calculate the raw scores, which corresponds to Z-scores. (a) 1.24 (b) 12 (c) 15.24 4. Let 𝑋~𝑁(300,25). Calculate the following probabilities. (a) 𝑃(𝑋 > 305) (b) 𝑃(𝑋 < 310) (c) 𝑃(𝑋 > 286) 5. Given 𝑋~𝑁(−8,12). Evaluate the following probabilities. (a) 𝑃(𝑋 < −5.12) (b) 𝑃(−11.23 < 𝑋 ≤ −4.23) 6. The bulbs produced by a factory are known to have life expectancy, which is normally distributed with a mean of 1000 hours and standard deviation of 10 hours. (a) Determine the probability that a bulb selected at random, will have life expectancy of (i) more than 1015 hours. (ii) less than 995 hours. (iii) between 998 hours and 1012 hours. (b) If two bulbs are selected at random, determine the probability that their life expectancy fall within 998 hours and 1012 hours. 7. The transmission on a model of specific car has a warranty for 40,000 km. It is known that the life of such a transmission has a normal distribution with a mean 72,000 km and a standard deviation of 12,000 km. Determine the percentage of the transmissions will (a) fail before the end of the warranty period (b) be good for more than 100,000 km Page 1 Copyright of Curtin University Tutorial 11.1 8. A manufacturer does not know that mean and standard deviation of the lengths of the iron nails he is producing. However, a control system rejects all nails longer than 2.5 cm and those under 2.0 cm in length. Out of 1000 nails produced, 10% are rejected as too short and 6% as too long. Calculate the mean and standard deviation of the iron nails produced. 9. A machine produces screws of variable diameter. Given that the diameter follows normal distribution with a mean of 10 mm and a variance of 4 mm2. A screw with diameter 8 mm and 12 mm are acceptable. Calculate (a) the percentage of the screws which is acceptable. (b) number of screws acceptable is a batch of 1000 screws. 10. A machine is set to fill packets of sugar with 200 g each. However, due to the inaccuracy of this type of the machine the actual weights in packets are normally distributed with mean of 201.5 g and a standard deviation of 4.8 g. A quality control was used by the factory to recycle any packet with less than 196 g of sugar. Calculate the following: (a) The percentage of the number of packets that will be recycled. (b) The number of packets which will be recycled if the factory produces 20,000 packets per day. (c) If a packet is selected from those destined for recycling, evaluate the probability that it is less than 191 g. (d) If the company wishes to reduce the percentage recycled to 5% by increasing the mean weight without affecting the standard deviation, calculate the mean weight should they set. Page 2  Tutorial 11 Topic: Probability Distribution II 1. The continuous random variable X has the probability density function shown below. f (x ) k x 2 4 6 8 10 Determine: (a) k (b) P(X < 4) (c) P(X < 8) (d) P(X > 4 | X < 8) 2. The daily trading volumes (millions of shares) for stocks traded on the New York Stock Exchange for 12 days in August and September are shown here: 917 938 1046 944 723 783 813 1057 766 836 992 973 The probability distribution of trading volume is approximately normal. (a) Compute the mean and standard deviation for the daily trading volume to use as estimates of the population mean and standard deviation. (b) What is the probability that on a particular day the trading volume will be less than 800 million shares? (c) What is the probability that trading volume will exceed 1 billion shares? 3. The masses of chickens sold at a market are normally distributed with mean 1.5 kg and standard deviation 400 g. (a) Determine the probability that a chicken, chosen at random, has a mass of more than 1.2 kg. (b) In one day 200 chickens are sold. Estimate how many chickens weigh less than 1.6 kg. 4. GMAT (Graduate Management Admissions Test) scores are normally distributed with a mean of 500 and a standard deviation of 100. (a) What percentage of scores fall in the interval from 350 to 450? (b) What percentage of scores fall in the interval from 550 to 650? (c) A certain graduate business school automatically accepts all applicants whose GMAT scores exceed 620. Approximately what percentage of all those taking the GMAT would qualify for admission to this school? Page 1 Tutorial 11 Page 2