TOPICAL REVISION PACKAGE III (SUGGESTED SOLUTIONS)
|
|
- Miles Price
- 5 years ago
- Views:
Transcription
1 Topical Revision Package III Suggested Solutions H Mathematics (7) PREPARATORY COURSE FOR SINGAPORE-CAMBRIDGE GENERAL CERTIFICATE OF EDUCATION (ADVANCED LEVEL) H MATHEMATICS (7) TOPICAL REVISION PACKAGE III (SUGGESTED SOLUTIONS) Version Page of
2 Topical Revision Package III Suggested Solutions H Mathematics (7) Permutations and Combinations ) JJC/II/ (i) ENDANGERED: E, N, D, A, G, R No. of -letter code-words from E, N, D, A, G, R C! 0 (ii) Select letter from N, D, A, G,R No. of -letter code-words with the chosen letter and E s (iii) Case I: same letters No. of such code-words = 0 Case II: pair of same letters! No. of code-words C C 0! Case III: pairs of same letters No. of code-words!!! C!! C 0 No. of -letter code-words that contain at least repeated letter = = ) MI/II/ (a) (i) Consider the green, yellow and purple tiles as unit. Number of ways to arrange the green, yellow and purple tiles within the unit!! Arranging the unit with the rest of the tiles Thus, no. of possible arrangement for the tiles 7!!!! 7! 00!!! Page of
3 Topical Revision Package III Suggested Solutions H Mathematics (7) (ii) Number of ways to arrange the red, green, yellow and purple tile Number of ways to slot in the blue tiles C!!! Number of possible arrangements such that no blue tiles are placed next to! another C 00!! (iii) Number of possible arrangements such that a red tile at the beginning and! another red tile at the end of the line 00.!! (b) (i) Number of ways = ( )! = 0 (ii) Insertion method Number of ways 7 7! C!! 00 (iii) Number of ways = ( )!! = 000 ) SRJC/II/ (a) Number of ways = = (b) (i) Number of parking arrangements (ii) =! (cases where the two cars are taking the side with lots) (cases where the two cars are taking the side with lots! C!!! C!!! Number of parking arrangements C!7! 00 ) DHS/II/ (i) Number of ways =! = 00 (ii) Number of ways to arrange the boys and girls on each side = Number of ways the group of boys and girls can arrange themselves =!! Required number of ways =!! = 0 (iii) Number of ways to choose any out of the seats to sit the boys and girls Number of ways to arrange the boys and girls! Required number of ways C! 00 C Page of
4 Topical Revision Package III Suggested Solutions H Mathematics (7) (iv) Number of ways to choose any out of the boys to sit on the seats and number of ways to arrange the boys C! Number of ways to choose any out of the seats to sit the remaining boys/ girls and number of ways to arrange the boys/ girls C! Required number of ways C! C! ) PJC/II/ Case : No. of code-words Case : No. of code-words Case : No. of code-words all letters are different C! 70 pair of repeated letters are used 7! C C 00! pairs of repeated letters are used! C C 0!! Total number of -letter code-words = = 0 ) RI/II/ (i) Number of ways for all man to stand next to his wife =! Required number of ways =!! = 70 (ii) Number of ways to choose the couple who don t stand next to each other C Number of ways to arrange the couples who are standing next to their spouse!! Number of ways to arrange each couple standing together =! = Number of ways to slot the remaining couple in between any two person P Required number of ways C! P Page of
5 Topical Revision Package III Suggested Solutions H Mathematics (7) 7) NYJC/II/ Number of ways = 7! + 7!! = 0 (i) Number of ways (ii) Number of ways!! 00!! 00! ) TJC/II/7 (a) (i) Number of ways C C C 0 (ii) There will be = empty seats Therefore, the number of ways of rearranging the three groups where each family of three sit together is!!!!! (b) Number of ways!! P 0 ) RVHS/II/7 (a) COORDINATION: O, I, N, A, C, R, D, T There are only possible arrangement format: (i) Consonant, Vowel, Consonant, Vowel,, Consonant, Vowel, Consonant, Vowel. (ii) Vowel, Consonant, Vowel, Consonant,, Vowel, Consonant, Vowel, Consonant. As there are Ns, O and Is, the total number of different arrangements in which the!! vowels and consonants alternates is 00.!!! (b) Case : consonants and vowels No. of possible passwords C C Case : consonants and vowel No. of possible passwords! 00 C C! Total no. of passwords with more than consonants = 00 + =. Page of
6 Topical Revision Package III Suggested Solutions H Mathematics (7) ) AJC/II/ (i) Family, Holiday, Friends 7 Choose images at least images from the Friends folder and at least image from each of the other two folders. Case : Friends + Holiday + Family No. of selections 7 C C C 0 Case : Friends + Holiday + Family No. of selections 7 C C C 0 Case : Friends + Holiday + Family No. of selections 7 C C C 0 Total no. of selections = = 70 (ii) Case : identical images in first row No. of arrangements =! Case : identical images in second row of No. of arrangements C! 0 Total no. of arrangements = = 00 (iii) Pimage A will appear a second time in the next n images 0. n 0. n 0. n 0. ln 0. n.7 ln The least value of n is. Page of
7 Topical Revision Package III Suggested Solutions H Mathematics (7) Probability ) CJC/II/7 (i) Pchosen serviceman is an Officer 0.0 (ii) 0.7[(0.0)(0.) + (0.)(0.7) + (0.7)(0.)] + (0.)(a)(0.) = a = 0.0 a = 0.0 Probability required P Failed NPFA an Officer P an Officer P Failed NPFA and an Officer P an Officer ( s.f.) 0.0 (iii) Probability required = = (iv) Probability Required P Failed NPFA a Non-Officer P a Non-Officer Pan Officer P Failed NPFA and an Officer ( s.f.) ( s.f.) ) SRJC/II/7 (a) (i) P A P A B P A B' P A P A B, A and B are independent events. AB P B Since (ii) P P B PB P A B P A PB P A B 7 Page 7 of
8 Topical Revision Package III Suggested Solutions H Mathematics (7) (b) (c) Required probability = (0.)(0.) + 0.7(0.)(0.7)(0.(0.) + Required probability = 0.[ + (0.7 0.) + (0.7 0.) + ] ! 7! C!! ) JJC/II/7 (i) For a set match, Probability of A winning = probability of B winning p(0.)(0.) + ( p)(0.)(0.7) = p(0.)(0.) + ( p)(0.)(0.) 0.p p = 0.p p 0.p = 0. p (Shown) (ii) P B wins the first set A wins the match P B wins the first set Awins the match P A wins the match = ( s.f.) Page of
9 Topical Revision Package III Suggested Solutions H Mathematics (7) ) IJC/II/ (i) P(the result is positive) = 0.7(0.) + 0.7(0.0) = ( s.f.) (ii) P(elderly has diabetes given that the result is positive) P elderly has diabetes and the result is positive P the result is positive ( s.f.) ) PJC/II/7 Let event A be watched Jogging man Let event B be watched Voice of me Given: P(A) = 0.7, P(B) = 0., P(A B ) = 0. P A B' (i) P AB' 0. P B' P A B' A B A A B (ii) P A' B P A B' PB P A B P A B' P P P ' = (0. 0.) + 0. = 0. P B A' 0.0 P BA' 0. P A' 0. (iii) Since P(A) = 0.7 P(A B ) = 0., the two events are not independent. Page of
10 Topical Revision Package III Suggested Solutions H Mathematics (7) ) ACJC/II/ (i) Required probability n n n n n n nn (ii) Required probability n n n nn n 7) NJC/II/ (a) (b) (c) Required probability P(one gets the flu) = 0.p + 0.(0.) C C ( s.f.) 0 C = 0.(0.) + (0.)(0.) = 0. or 0 ( s.f.) P(exactly one of the two people took the flu vaccine both of them get the flu) ! 0. = ( s.f.) ) RI/II/ Let A and B be the event that a student takes Mathematics A and Mathematics B respectively. Let F be the event that a student failed the paper that he/she sat for. Page of
11 Topical Revision Package III Suggested Solutions H Mathematics (7) 0.p 0 p 0 0.p P p 0 0.p P F ' 0 0 P(exactly one out of failed) = P(one failed and one passed) 0 0.p 0 0.p C p p 0.0p 000 = 0.000(00 p 0.0p ) (Shown) P(both take Mathematics A exactly one out of failed) (i) F (ii) P one takes Mathematics Aand failed and the other takes Mathematics Aand passed P exactly one out of failed 0.p 0.7 p C p0.0 p 0.p 7 00 p0.0p Hence,.0p = 00 p 0.07p.p + p 00 = 0 From the G.C., p = 0 or 0 (rej since p 0) ) AJC/II/7 AB P A B' P B' (i) P ' P A B' PB Thus, P A B P A P A B' 0 Page of
12 Topical Revision Package III Suggested Solutions H Mathematics (7) (ii) P A' AB P P A' B A B PB P AB A B A B P P P 0 0 (iii) A is independent of C P A' C P A' PC (iv) P(A B C) is greatest when A C is totally in B. P A ' C P A' B C P(A B C) is least when A C is furthest away from B. Since P A' B' P A' B C 0. (Shown) 0 P A' B C ) DHS/II/7 (i) There are! ways to arrange the girls. There are possible spaces where boys can be slotted in. Choose spaces out of. There are! ways to arrange the boys. P all boys are separated! C!! or 0. ( s.f.) Page of
13 Topical Revision Package III Suggested Solutions H Mathematics (7) (ii) Choose out of girls to be on both sides of the particular boy. There are! ways to arrange the girls. Taking GBG as one group, there are! ways to arrange the group and other students. P a particular boy is between girls C!!! or 0. ( s.f.) (iii) Taking the boys as one group, there are! ways to arrange the boys. Taking the boys group as reference point, there are! ways to arrange the rest of the girls. P all boys next to one another in a circle (iv) Pat least student from each race!!! P Chinese and Malaly only or Chinese and Indian only C C 7 7 C7 or 0.0 ( s.f.) or 0. ( s.f.) Page of
14 Topical Revision Package III Suggested Solutions H Mathematics (7) Discrete Random Variable P p0 p p --- (). Since X x all w Given that E X 0 p p p 0p0 p p --- () 0 Given that Var X E X 0 p0 p p p p p --- () 0 From the G.C., solving (), () and (), we get p0, p and 0 p. The probability distribution table for R is r 0 P(R = r) ER r PR r 0 all r Var R E R 0 0 Page of
15 Topical Revision Package III Suggested Solutions H Mathematics (7). P (a) Since X x all x (Shown) E 0 (b) X or. 0 E X X X X Var E E..7 (c) The possible pairs are (, ), (,), (,), (, ). Required probability 7 or 0. ( s.f.). Note: PY 0 PY PY Hence, the probability distribution of Y is Y 0 P(Y = y) Then, EY y PY y 0 all y (Shown) Note: ET EY y PY y 0 all y Var Var E 0 T Y Y (Shown) Page of
16 Topical Revision Package III Suggested Solutions H Mathematics (7). (i) The table of outcome is Disc Die Note that the probabilty to get any value in the table of outcome is Hence, the probability distribution table of X is. (ii) P X X 7 r 7 P(R = r) P X X 7 P X 7 X X X X P X P P P P 7 (Shown) (iii) E X (since the distribution is symmetrical about X = ) Var X E X 7 (Shown) (iv) Var X X Var X Var X Var X (v) P XX P X 7P X P X P X P X P X Page of
17 Topical Revision Package III Suggested Solutions H Mathematics (7) P X x p + q =. (a) Note: (b) all x p + q = () Given E(X) =. (0.) + (0.) + p + q =. p + q = () From the G.C., solving () and (), we get p = 0. and q = 0.0. Var(X) = E[(X μ) ] all x x P X x = (.) (0.) + (.) (0.) + (.) (0.) + (.) (0.0) =. Let Y be the amount of money received by a player in on roll. The probability distribution of Y is y 0 P(Y = y) Note: EY Let n be the number of rolls a player has. We want E(Y + Y + + Y n ) = E(nY) = ne(y) = n n The player should be allowed rolls to receive $ Page 7 of
18 Topical Revision Package III Suggested Solutions H Mathematics (7) 7. (a) Red Ball drawn Non-Red Ball drawn Red Ball drawn Non-Red Ball drawn Red Ball drawn Non-Red Ball drawn Red Ball drawn Let R be the amount of points awarded. Hence, from the probability tree diagram above, the probability distribution table of R is r 7 P(R = r) ER r PR r 7 all r (b) From the question, the values X can take are 0,,,,, 0 and. Consider the table of outcome for X: To get : (, ), (, ), (, ), (,) To get : (, ), (, ), (, ), (,) To get : (, ), (, ) To get : (, ), (, ) To get 0: (, ), (, ) To get : (, ) Thus, we have the following probability distribution table: Hence, E x 0 0 P(X = x) X x P X x all x Page of
19 Topical Revision Package III Suggested Solutions H Mathematics (7) E X x P X x all x X X X Var E E 7 7. Note: R can only take the values 0, and since there are only red balls in the bottle. 7 7 PR PR PR r P(R = r) 7 PR PR PR (Shown) 7 7 ER r PR r 0 all r (Shown) 7 7 Var R E R 0 7 Note: M = R 7 7 E P all j Note: J j J j j P(J = j) 7 7 Page of
20 Topical Revision Package III Suggested Solutions H Mathematics (7) Thus, 7 7 Var J E J 7 EM J EM EJ ER EJ 0 Var Var Var Var Var M J M J R J E X xp X x all x E X x P X x all x 7 Thus, Var X E X E X Let Y = X X. 7 (Shown) y 0 P(Y = y) E P 0 all y Thus, Y y Y y (a) (b) True. Since W values is the same as the Y, their respective expectation will be the same. False. Since there are more W values than Y values, their respective variance will not be the same. Page 0 of
21 Topical Revision Package III Suggested Solutions H Mathematics (7). Let Y denote the score of the first die when thrown once. E P all y Y y Y y Y Y Var E Let W denote the score of the second die when thrown once. E P all w W w W w For variance, since Y EY W EW, then E Y EY E W EW. Thus, Var W Var Y. 7 Note: X = Y + Y + W, where Y and Y are independent observations of Y. (a) Var X Var Y Var Y Var W Var Y (Shown) 7 7 E E E E E E (b) Note: X Y Y W Y W 7 P P X Thus, X 7 7 P X P X 7 7 P X P X X X P.0 P. X X P P Page of
22 Topical Revision Package III Suggested Solutions H Mathematics (7) (Shown) Page of
Topic 5: Probability. 5.4 Combined Events and Conditional Probability Paper 1
Topic 5: Probability Standard Level 5.4 Combined Events and Conditional Probability Paper 1 1. In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music. The Venn
More informationMath SL Day 66 Probability Practice [196 marks]
Math SL Day 66 Probability Practice [96 marks] Events A and B are independent with P(A B) = 0.2 and P(A B) = 0.6. a. Find P(B). valid interpretation (may be seen on a Venn diagram) P(A B) + P(A B), 0.2
More informationWeek 04 Discussion. a) What is the probability that of those selected for the in-depth interview 4 liked the new flavor and 1 did not?
STAT Wee Discussion Fall 7. A new flavor of toothpaste has been developed. It was tested by a group of people. Nine of the group said they lied the new flavor, and the remaining 6 indicated they did not.
More informationOCR Statistics 1 Probability. Section 1: Introducing probability
OCR Statistics Probability Section : Introducing probability Notes and Examples These notes contain subsections on Notation Sample space diagrams The complement of an event Mutually exclusive events Probability
More informationNotes for Math 324, Part 17
126 Notes for Math 324, Part 17 Chapter 17 Common discrete distributions 17.1 Binomial Consider an experiment consisting by a series of trials. The only possible outcomes of the trials are success and
More informationDiscussion 03 Solutions
STAT Discussion Solutions Spring 8. A new flavor of toothpaste has been developed. It was tested by a group of people. Nine of the group said they liked the new flavor, and the remaining indicated they
More informationSolutionbank S1 Edexcel AS and A Level Modular Mathematics
Heinemann Solutionbank: Statistics S Page of Solutionbank S Exercise A, Question Write down whether or not each of the following is a discrete random variable. Give a reason for your answer. a The average
More informationProblems and results for the ninth week Mathematics A3 for Civil Engineering students
Problems and results for the ninth week Mathematics A3 for Civil Engineering students. Production line I of a factor works 0% of time, while production line II works 70% of time, independentl of each other.
More informationNuevo examen - 02 de Febrero de 2017 [280 marks]
Nuevo examen - 0 de Febrero de 0 [0 marks] Jar A contains three red marbles and five green marbles. Two marbles are drawn from the jar, one after the other, without replacement. a. Find the probability
More informationH2 Mathematics Probability ( )
H2 Mathematics Probability (208 209) Practice Questions. For events A and B it is given that P(A) 0.7, P(B) 0. and P(A B 0 )0.8. Find (i) P(A \ B 0 ), [2] (ii) P(A [ B), [2] (iii) P(B 0 A). [2] For a third
More information4/17/2012. NE ( ) # of ways an event can happen NS ( ) # of events in the sample space
I. Vocabulary: A. Outcomes: the things that can happen in a probability experiment B. Sample Space (S): all possible outcomes C. Event (E): one outcome D. Probability of an Event (P(E)): the likelihood
More informationConditional Probability
Conditional Probability Terminology: The probability of an event occurring, given that another event has already occurred. P A B = ( ) () P A B : The probability of A given B. Consider the following table:
More informationWhen working with probabilities we often perform more than one event in a sequence - this is called a compound probability.
+ Independence + Compound Events When working with probabilities we often perform more than one event in a sequence - this is called a compound probability. Compound probabilities are more complex than
More informationIdentify events like OIL+ in the tree and their probabilities (don't reduce).
1. Make a complete tree diagram for the following information: P(OIL) = 0.3 P(+ OIL) = 0.8 P(- no OIL) = 0.9 Identify events like OIL+ in the tree and their probabilities (don't reduce). 2. From your tree
More informationSTAT 516 Answers Homework 2 January 23, 2008 Solutions by Mark Daniel Ward PROBLEMS. = {(a 1, a 2,...) : a i < 6 for all i}
STAT 56 Answers Homework 2 January 23, 2008 Solutions by Mark Daniel Ward PROBLEMS 2. We note that E n consists of rolls that end in 6, namely, experiments of the form (a, a 2,...,a n, 6 for n and a i
More informationSTA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru Venkateswara Rao, Ph.D. STA 2023 Fall 2016 Venkat Mu ALL THE CONTENT IN THESE SOLUTIONS PRESENTED IN BLUE AND BLACK
More informationChapter 2 PROBABILITY SAMPLE SPACE
Chapter 2 PROBABILITY Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation,
More informationSTEP Support Programme. Statistics STEP Questions: Solutions
STEP Support Programme Statistics STEP Questions: Solutions 200 S Q2 Preparation (i) (a) The sum of the probabilities is, so we have k + 2k + 3k + 4k k 0. (b) P(X 3) P(X 3) + P(X 4) 7 0. (c) E(X) 0 ( +
More informationF71SM STATISTICAL METHODS
F71SM STATISTICAL METHODS RJG SUMMARY NOTES 2 PROBABILITY 2.1 Introduction A random experiment is an experiment which is repeatable under identical conditions, and for which, at each repetition, the outcome
More informationSTA 2023 EXAM-2 Practice Problems. Ven Mudunuru. From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru, Venkateswara Rao STA 2023 Spring 2016 1 1. A committee of 5 persons is to be formed from 6 men and 4 women. What
More informationHW MATH425/525 Lecture Notes 1
HW MATH425/525 Lecture Notes 1 Definition 4.1 If an experiment can be repeated under the same condition, its outcome cannot be predicted with certainty, and the collection of its every possible outcome
More informationPhysicsAndMathsTutor.com
1. Philip and James are racing car drivers. Philip s lap times, in seconds, are normally distributed with mean 90 and variance 9. James lap times, in seconds, are normally distributed with mean 91 and
More informationUC Berkeley, CS 174: Combinatorics and Discrete Probability (Fall 2008) Midterm 1. October 7, 2008
UC Berkeley, CS 74: Combinatorics and Discrete Probability (Fall 2008) Midterm Instructor: Prof. Yun S. Song October 7, 2008 Your Name : Student ID# : Read these instructions carefully:. This is a closed-book
More informationCHAPTER - 16 PROBABILITY Random Experiment : If an experiment has more than one possible out come and it is not possible to predict the outcome in advance then experiment is called random experiment. Sample
More informationEdexcel past paper questions
Edexcel past paper questions Statistics 1 Discrete Random Variables Past examination questions Discrete Random variables Page 1 Discrete random variables Discrete Random variables Page 2 Discrete Random
More informationDiscrete Random Variable Practice
IB Math High Level Year Discrete Probability Distributions - MarkScheme Discrete Random Variable Practice. A biased die with four faces is used in a game. A player pays 0 counters to roll the die. The
More informationSome Basic Concepts of Probability and Information Theory: Pt. 1
Some Basic Concepts of Probability and Information Theory: Pt. 1 PHYS 476Q - Southern Illinois University January 18, 2018 PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and
More informationUNIT Explain about the partition of a sampling space theorem?
UNIT -1 1. Explain about the partition of a sampling space theorem? PARTITIONS OF A SAMPLE SPACE The events B1, B2. B K represent a partition of the sample space 'S" if (a) So, when the experiment E is
More informationIB Math Standard Level Probability Practice 2 Probability Practice 2 (Discrete& Continuous Distributions)
IB Math Standard Level Probability Practice Probability Practice (Discrete& Continuous Distributions). A box contains 5 red discs and 5 black discs. A disc is selected at random and its colour noted. The
More informationSolutions - Final Exam
Solutions - Final Exam Instructors: Dr. A. Grine and Dr. A. Ben Ghorbal Sections: 170, 171, 172, 173 Total Marks Exercise 1 7 Exercise 2 6 Exercise 3 6 Exercise 4 6 Exercise 5 6 Exercise 6 9 Total 40 Score
More informationMidterm Exam 1 Solution
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2015 Kannan Ramchandran September 22, 2015 Midterm Exam 1 Solution Last name First name SID Name of student on your left:
More informationCMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 8 Prof. Hanna Wallach wallach@cs.umass.edu February 16, 2012 Reminders Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
More information324 Stat Lecture Notes (1) Probability
324 Stat Lecture Notes 1 robability Chapter 2 of the book pg 35-71 1 Definitions: Sample Space: Is the set of all possible outcomes of a statistical experiment, which is denoted by the symbol S Notes:
More informationMTH302 Long Solved Questions By
MTH30 Long Solved uestions By www.vuattach.ning.com If you toss a die and observe the number of dots that appears on top face then write the events that the even number occurs. Number of Possible outcomes
More informationChapter 8 Sequences, Series, and Probability
Chapter 8 Sequences, Series, and Probability Overview 8.1 Sequences and Series 8.2 Arithmetic Sequences and Partial Sums 8.3 Geometric Sequences and Partial Sums 8.5 The Binomial Theorem 8.6 Counting Principles
More informationDept. of Linguistics, Indiana University Fall 2015
L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would
More informationMath 447. Introduction to Probability and Statistics I. Fall 1998.
Math 447. Introduction to Probability and Statistics I. Fall 1998. Schedule: M. W. F.: 08:00-09:30 am. SW 323 Textbook: Introduction to Mathematical Statistics by R. V. Hogg and A. T. Craig, 1995, Fifth
More informationMath 218 Supplemental Instruction Spring 2008 Final Review Part A
Spring 2008 Final Review Part A SI leaders: Mario Panak, Jackie Hu, Christina Tasooji Chapters 3, 4, and 5 Topics Covered: General probability (probability laws, conditional, joint probabilities, independence)
More information$ and det A = 14, find the possible values of p. 1. If A =! # Use your graph to answer parts (i) (iii) below, Working:
& 2 p 3 1. If A =! # $ and det A = 14, find the possible values of p. % 4 p p" Use your graph to answer parts (i) (iii) below, (i) Find an estimate for the median score. (ii) Candidates who scored less
More information6.041/6.431 Spring 2009 Quiz 1 Wednesday, March 11, 7:30-9:30 PM. SOLUTIONS
6.0/6.3 Spring 009 Quiz Wednesday, March, 7:30-9:30 PM. SOLUTIONS Name: Recitation Instructor: Question Part Score Out of 0 all 0 a 5 b c 5 d 5 e 5 f 5 3 a b c d 5 e 5 f 5 g 5 h 5 Total 00 Write your solutions
More informationIntro to Probability Day 4 (Compound events & their probabilities)
Intro to Probability Day 4 (Compound events & their probabilities) Compound Events Let A, and B be two event. Then we can define 3 new events as follows: 1) A or B (also A B ) is the list of all outcomes
More informationTopic 5 Part 3 [257 marks]
Topic 5 Part 3 [257 marks] Let 0 3 A = ( ) and 2 4 4 0 B = ( ). 5 1 1a. AB. 1b. Given that X 2A = B, find X. The following table shows the probability distribution of a discrete random variable X. 2a.
More informationWhat is Probability? Probability. Sample Spaces and Events. Simple Event
What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5
More informationHomework 4 Solution, due July 23
Homework 4 Solution, due July 23 Random Variables Problem 1. Let X be the random number on a die: from 1 to. (i) What is the distribution of X? (ii) Calculate EX. (iii) Calculate EX 2. (iv) Calculate Var
More informationStatistics for Business and Economics
Statistics for Business and Economics Basic Probability Learning Objectives In this lecture(s), you learn: Basic probability concepts Conditional probability To use Bayes Theorem to revise probabilities
More informationDiscrete Mathematics and Probability Theory Fall 2011 Rao Midterm 2 Solutions
CS 70 Discrete Mathematics and Probability Theory Fall 20 Rao Midterm 2 Solutions True/False. [24 pts] Circle one of the provided answers please! No negative points will be assigned for incorrect answers.
More informationCPT Solved Scanner (English) : Appendix 71
CPT Solved Scanner (English) : Appendix 71 Paper-4: Quantitative Aptitude Chapter-1: Ratio and Proportion, Indices and Logarithm [1] (b) The integral part of a logarithms is called Characteristic and the
More information1 The Basic Counting Principles
1 The Basic Counting Principles The Multiplication Rule If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n ways [regardless of how
More informationSALES AND MARKETING Department MATHEMATICS. Combinatorics and probabilities. SOLUTIONS of tutorials and exercises
SALS AND MARKTING Department MATHMATICS 2 nd Semester Combinatorics and probabilities SOLUTIONS of tutorials and exercises Online document : http://jff-dut-tc.weebly.com section DUT Maths S2 IUT de Saint-tienne
More informationIB Math High Level Year 1 Probability Practice 1
IB Math High Level Year Probability Practice Probability Practice. A bag contains red balls, blue balls and green balls. A ball is chosen at random from the bag and is not replaced. A second ball is chosen.
More informationPROBABILITY.
PROBABILITY PROBABILITY(Basic Terminology) Random Experiment: If in each trial of an experiment conducted under identical conditions, the outcome is not unique, but may be any one of the possible outcomes,
More informationISyE 6739 Test 1 Solutions Summer 2015
1 NAME ISyE 6739 Test 1 Solutions Summer 2015 This test is 100 minutes long. You are allowed one cheat sheet. 1. (50 points) Short-Answer Questions (a) What is any subset of the sample space called? Solution:
More informationClass 26: review for final exam 18.05, Spring 2014
Probability Class 26: review for final eam 8.05, Spring 204 Counting Sets Inclusion-eclusion principle Rule of product (multiplication rule) Permutation and combinations Basics Outcome, sample space, event
More informationTheoretical Probability (pp. 1 of 6)
Theoretical Probability (pp. 1 of 6) WHAT ARE THE CHANCES? Objectives: Investigate characteristics and laws of probability. Materials: Coin, six-sided die, four-color spinner divided into equal sections
More informationProbability 5-4 The Multiplication Rules and Conditional Probability
Outline Lecture 8 5-1 Introduction 5-2 Sample Spaces and 5-3 The Addition Rules for 5-4 The Multiplication Rules and Conditional 5-11 Introduction 5-11 Introduction as a general concept can be defined
More informationProbability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin,
Chapter 8 Exercises Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin, morin@physics.harvard.edu 8.1 Chapter 1 Section 1.2: Permutations 1. Assigning seats *
More information[ 1] ST301(AKI) Mid1 2010/10/07. ST 301 (AKI) Mid 1 PLEASE DO NOT OPEN YET! TURN OFF YOUR CELL PHONE! FIRST NAME: LAST NAME: STUDENT ID:
[ 1] ST301(AKI) Mid1 2010/10/07 ST 301 (AKI) Mid 1 PLEASE DO NOT OPEN YET! TURN OFF YOUR CELL PHONE! FIRST NAME: LAST NAME: STUDENT ID: [ 2] ST301(AKI) Mid1 2010/10/07 Choose one answer for each question.
More informationProbabilistic models
Probabilistic models Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became
More information3 PROBABILITY TOPICS
Chapter 3 Probability Topics 135 3 PROBABILITY TOPICS Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr) Introduction It is often necessary
More informationSTAT 201 Chapter 5. Probability
STAT 201 Chapter 5 Probability 1 2 Introduction to Probability Probability The way we quantify uncertainty. Subjective Probability A probability derived from an individual's personal judgment about whether
More informationUpdated Jan SESSION 4 Permutations Combinations Polynomials
SESSION 4 Permutations Combinations Polynomials Mathematics 30-1 Learning Outcomes Permutations and Combinations General Outcome: Develop algebraic and numeric reasoning that involves combinatorics. Specific
More informationConditional probability
CHAPTER 4 Conditional probability 4.1. Introduction Suppose there are 200 men, of which 100 are smokers, and 100 women, of which 20 are smokers. What is the probability that a person chosen at random will
More information384 PU M.Sc Five year Integrated M.Sc Programmed (Mathematics, Computer Science,Statistics)
384 PU M.Sc Five year Integrated M.Sc Programmed (Mathematics, Computer Science,Statistics) 1 of 1 146 PU_216_384_E 2 cos 1 π 4 cos 1 cos1 2 of 1 15 PU_216_384_E 1! 1 1 3 of 1 15 PU_216_384_E 1 4 of 1
More information, x {1, 2, k}, where k > 0. Find E(X). (2) (Total 7 marks)
1.) The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). Show that k = 3. (1) Find E(X). (Total 7 marks) 2.) In a group
More informationSteve Smith Tuition: Maths Notes
Maths Notes : Discrete Random Variables Version. Steve Smith Tuition: Maths Notes e iπ + = 0 a + b = c z n+ = z n + c V E + F = Discrete Random Variables Contents Intro The Distribution of Probabilities
More informationCOVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS
COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS COURSE: CBS 221 DISCLAIMER The contents of this document are intended for practice and leaning purposes at the undergraduate
More informationCS 109 Midterm Review!
CS 109 Midterm Review! Major Topics: Counting and Combinatorics Probability Conditional Probability Random Variables Discrete/Continuous Distributions Joint Distributions and Convolutions Counting Sum
More informationWhat is a random variable
OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE MATH 256 Probability and Random Processes 04 Random Variables Fall 20 Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr
More information3/15/2010 ENGR 200. Counting
ENGR 200 Counting 1 Are these events conditionally independent? Blue coin: P(H = 0.99 Red coin: P(H = 0.01 Pick a random coin, toss it twice. H1 = { 1 st toss is heads } H2 = { 2 nd toss is heads } given
More informationDescriptive Statistics and Probability Test Review Test on May 4/5
Descriptive Statistics and Probability Test Review Test on May 4/5 1. The following frequency distribution of marks has mean 4.5. Mark 1 2 3 4 5 6 7 Frequency 2 4 6 9 x 9 4 Find the value of x. Write down
More informationLesson One Hundred and Sixty-One Normal Distribution for some Resolution
STUDENT MANUAL ALGEBRA II / LESSON 161 Lesson One Hundred and Sixty-One Normal Distribution for some Resolution Today we re going to continue looking at data sets and how they can be represented in different
More information6.1 Logic. Statements or Propositions. Negation. The negation of a statement, p, is not p and is denoted by p Truth table: p p
6.1 Logic Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt
More informationSS257a Midterm Exam Monday Oct 27 th 2008, 6:30-9:30 PM Talbot College 342 and 343. You may use simple, non-programmable scientific calculators.
SS657a Midterm Exam, October 7 th 008 pg. SS57a Midterm Exam Monday Oct 7 th 008, 6:30-9:30 PM Talbot College 34 and 343 You may use simple, non-programmable scientific calculators. This exam has 5 questions
More informationSection 4.2 Basic Concepts of Probability
Section 4.2 Basic Concepts of Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 88 Section 4.2 Objectives Identify the sample space of a probability experiment Identify simple events Use
More informationProbability and Statistics Notes
Probability and Statistics Notes Chapter One Jesse Crawford Department of Mathematics Tarleton State University (Tarleton State University) Chapter One Notes 1 / 71 Outline 1 A Sketch of Probability and
More informationMATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3
MATH 250 / SPRING 2011 SAMPLE QUESTIONS / SET 3 1. A four engine plane can fly if at least two engines work. a) If the engines operate independently and each malfunctions with probability q, what is the
More informationIntroduction to Probability, Fall 2009
Introduction to Probability, Fall 2009 Math 30530 Review questions for exam 1 solutions 1. Let A, B and C be events. Some of the following statements are always true, and some are not. For those that are
More informationBusiness Statistics MBA Pokhara University
Business Statistics MBA Pokhara University Chapter 3 Basic Probability Concept and Application Bijay Lal Pradhan, Ph.D. Review I. What s in last lecture? Descriptive Statistics Numerical Measures. Chapter
More informationSTEP Support Programme. Statistics STEP Questions
STEP Support Programme Statistics STEP Questions This is a selection of STEP I and STEP II questions. The specification is the same for both papers, with STEP II questions designed to be more difficult.
More informationSTAT/MA 416 Answers Homework 4 September 27, 2007 Solutions by Mark Daniel Ward PROBLEMS
STAT/MA 416 Answers Homework 4 September 27, 2007 Solutions by Mark Daniel Ward PROBLEMS 2. We ust examine the 36 possible products of two dice. We see that 1/36 for i = 1, 9, 16, 25, 36 2/36 for i = 2,
More informationTerm Definition Example Random Phenomena
UNIT VI STUDY GUIDE Probabilities Course Learning Outcomes for Unit VI Upon completion of this unit, students should be able to: 1. Apply mathematical principles used in real-world situations. 1.1 Demonstrate
More informationCarleton University. Final Examination Winter DURATION: 2 HOURS No. of students: 275
Carleton University Final Examination Winter 2017 DURATION: 2 HOURS No. of students: 275 Department Name & Course Number: Computer Science COMP 2804B Course Instructor: Michiel Smid Authorized memoranda:
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationSTAT 302: Assignment 1
STAT 302: Assignment 1 Due date: Feb 4, 2011. Hand to Mailbox besides LSK333 1. An imaginary small university has 3 programs A, B and C. If an applicant is a girl, her probability of being admitted to
More information( ) 2 + ( 2 x ) 12 = 0, and explain why there is only one
IB Math SL Practice Problems - Algebra Alei - Desert Academy 0- SL Practice Problems Algebra Name: Date: Block: Paper No Calculator. Consider the arithmetic sequence, 5, 8,,. (a) Find u0. (b) Find the
More informationMTH302 Quiz # 4. Solved By When a coin is tossed once, the probability of getting head is. Select correct option:
MTH302 Quiz # 4 Solved By konenuchiha@gmail.com When a coin is tossed once, the probability of getting head is. 0.55 0.52 0.50 (1/2) 0.51 Suppose the slope of regression line is 20 and the intercept is
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Lecture 2 Probability Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 3-1 3.1 Definition Random Experiment a process leading to an uncertain
More informationChapter 4 Probability
4-1 Review and Preview Chapter 4 Probability 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting
More informationCarleton University. Final Examination Winter DURATION: 2 HOURS No. of students: 152
Carleton University Final Examination Winter 2014 DURATION: 2 HOURS No. of students: 152 Department Name & Course Number: Computer Science COMP 2804B Course Instructor: Michiel Smid Authorized memoranda:
More informationPhysicsAndMathsTutor.com. Advanced/Advanced Subsidiary. You must have: Mathematical Formulae and Statistical Tables (Blue)
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Statistics S1 Advanced/Advanced Subsidiary Candidate Number Friday 20 January 2017 Afternoon Time: 1
More informationQ1 Own your learning with flash cards.
For this data set, find the mean, mode, median and inter-quartile range. 2, 5, 6, 4, 7, 4, 7, 2, 8, 9, 4, 11, 9, 9, 6 Q1 For this data set, find the sample variance and sample standard deviation. 89, 47,
More informationMassachusetts Institute of Technology
6041/6431: Probabilistic Systems Analysis Problem Set 3 Solutions Due September 9, 010 1 The hats of n persons are thrown into a box The persons then pic up their hats at random (ie, so that every assignment
More informationDiscrete Structures Prelim 1 Selected problems from past exams
Discrete Structures Prelim 1 CS2800 Selected problems from past exams 1. True or false (a) { } = (b) Every set is a subset of its power set (c) A set of n events are mutually independent if all pairs of
More informationExam 1 - Math Solutions
Exam 1 - Math 3200 - Solutions Spring 2013 1. Without actually expanding, find the coefficient of x y 2 z 3 in the expansion of (2x y z) 6. (A) 120 (B) 60 (C) 30 (D) 20 (E) 10 (F) 10 (G) 20 (H) 30 (I)
More informationMath 510 midterm 3 answers
Math 51 midterm 3 answers Problem 1 (1 pts) Suppose X and Y are independent exponential random variables both with parameter λ 1. Find the probability that Y < 7X. P (Y < 7X) 7x 7x f(x, y) dy dx e x e
More informationMATHEMATICS & STATISTICS TOPIC : CORRELATION DURATION 3 HR MARKS 80. Q1. (A) Attempt ANY SIX OF THE FOLLOWING (12) dy = 18x x log7 dx 2 x x
MATHEMATICS & STATISTICS MATRICES + FYJC FINAL EXAM 02 TOPIC : CORRELATION DURATION 3 HR MARKS 80 Q1. (A) Attempt ANY SIX OF THE FOLLOWING (12) 01. Differentiate the following function with respect to
More informationProducing data Toward statistical inference. Section 3.3
Producing data Toward statistical inference Section 3.3 Toward statistical inference Idea: Use sampling to understand statistical inference Statistical inference is when a conclusion about a population
More informationSTAT/MA 416 Answers Homework 6 November 15, 2007 Solutions by Mark Daniel Ward PROBLEMS
STAT/MA 4 Answers Homework November 5, 27 Solutions by Mark Daniel Ward PROBLEMS Chapter Problems 2a. The mass p, corresponds to neither of the first two balls being white, so p, 8 7 4/39. The mass p,
More informationOutline. Probability. Math 143. Department of Mathematics and Statistics Calvin College. Spring 2010
Outline Math 143 Department of Mathematics and Statistics Calvin College Spring 2010 Outline Outline 1 Review Basics Random Variables Mean, Variance and Standard Deviation of Random Variables 2 More Review
More informationgreen, green, green, green, green The favorable outcomes of the event are blue and red.
0 Chapter Review Review Key Vocabulary experiment, p. 0 outcomes, p. 0 event, p. 0 favorable outcomes, p. 0 probability, p. 08 relative frequency, p. Review Examples and Exercises experimental probability,
More information