STATPRO Exercises with Solutions. Problem Set A: Basic Probability
|
|
- Matilda Nichols
- 5 years ago
- Views:
Transcription
1 Problem Set A: Basic Probability 1. A tea taster is required to taste and rank three varieties of tea namely Tea A, B and C; according to the tasters preference. (ranking the teas from the best choice until the tasters least choice) a) Define the experiment Ranking the three varieties of tea from the tasters preference (e.g. first, second, last choice) b) List the sample points in the sample space. (Tasters order of preference: first choice, second choice and last choice) A B C B C A A C B C A B B A C C B A c) If the taster had no ability to distinguish a difference in the taste between teas, what is the probability that the taster will rank Tea A as best? Let A be the event that the taster ranked Tea A as the best. P(A) = n N = 2 6 = 1 3 d) Using the same condition in c) what is the probability that the taster will rank Tea A as least desirable? Let B be the event that the taster ranked Tea A as the least desirable. P(B) = n N = 2 6 = Two dice are tossed. What is the probability that the sum of the numbers showing on the dice is equal to 9? Prepared by Dr. Francis Joseph H. Campeña 1
2 Let C be the event that the sum of the numbers showing on the dice is equal to 9. The following shows the possible outcomes of event C when a pair of dice is tossed. P(C) = n N = 4 36 = In how many different ways can a true-false test consisting of questions be answered? We use the fundamental principle of counting in finding the number of ways of answering the test. Two factorial ways of answering question number 1, two factorial ways of answering question number 2 and so on (2!)(2!)(2!)(2!)(2!)(2!)(2!)(2!)(2!)(2!) = 2 = 1, How many distinct permutations can be made to the letters of the word columns. Since all the letters in the word columns are all distinct we can just use the formula n! to determine the number of distinct permutations of the letters in the word columns. n! = (7)! = 5, A college plays 12 football games during a season. In how many ways can the team end the season with 7 wins, 3 losses and 2 ties? 12C 7 5 C 3 2 C 2 Selecting a game where the team won Selecting a game where the team lost Selecting a game where the team tied to another team 6. From a group of 4 men and 5 women, how many committees of size 3 are possible a. With no restrictions? 9! n = 9 C 3 = (9 3)! 3! = 9! 6! 3! = 84 Prepared by Dr. Francis Joseph H. Campeña 2
3 b. With 1 man and 2 women? n = 4 C 1 5 C 2 = 4 = 40 Selecting a man Selecting a woman c. With 2 men and 1 woman if a certain man must be on the committee? If a certain man must be in the committee, we can now only chose one man and a woman to form the committee. n = 3 C 1 5 C 1 = 3 5 = 15 Selecting a man Selecting a woman Problem Set B: Laws on Probability 1. A certain genetic characteristic occurs in mice with probability equal to 0.2. If two mice are randomly selected from a large number of unrelated litters, what is the probability that both mice possess the genetic characteristic? Suppose events A and B are events of selecting the first and second mouse having a certain genetic characteristic. Since the mice are randomly selected, and the probability that the second mouse has the genetic characteristics does not depend on the probability that the first mouse also has the genetic characteristic we have the following: P(A B) = P(A) P(B) = (0.2)(0.2) = A lineup of men is conducted to test the ability of a witness to identify three burglary suspects. Suppose that the three burglary suspects who committed the crime are in the lineup. If the witness is actually unable to identify the suspects but feels compelled to make a choice, a) What is the probability that the three guilty men are selected by chance? Let A be the event that the three guilty men are selected by the witness. P(A) = n = 3 C 3 = 1 N C b) What is the probability that the witness selects three innocent men? Let B be the event that the three innocent men are selected by the witness. P(B) = n = 7 C 3 = 35 = 7 N C Prepared by Dr. Francis Joseph H. Campeña 3
4 3. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems and a dictionary, what is the probability that a) The dictionary is selected? Let B be the event that the dictionary is selected. P(A) = n = 8 C 2 1 C 1 = 28 = 1 N 9C b) 2 novels and 1 book of poems are selected? Let B be the event that the dictionary is selected. P(A) = n = 5 C 2 3 C 1 = 30 = 5 N 9C In a college graduating class of 0 students, 54 studied mathematics, 69 studied history and 35 studied both mathematics and history. If one of these students is selected at random, find the probability that Let A be the event that the student selected studied mathematics B be the event that the student selected studied history P(A) = 54 0 P(B) = 69 0 P(A B) = 35 0 a) The student takes mathematics or history P(A B) = P(A) + P(B) P(A B) = = 88 0 b) The student does not take either of these subjects The event (A B) c is the event where the student does not take either the math or history subjects. P(A B) + P((A B) c ) = P((A B)c ) = 1 P((A B) c ) = = 12 0 c) The student takes history but not mathematics The event A c B is the event where student takes history but not mathematics - = Prepared by Dr. Francis Joseph H. Campeña 4
5 Thus we have the following: P(A c B ) = P(A) P(A B) = = The probability that a married man watches a telenovelas is 0.4 and the probability that a married woman watches the show is 0.8. The probability that a man watches the show given that his wife does is Find the probability that Let M be the event that a married man watches telenovelas is selected W be the event that a married woman watches telenovelas is selected P(M) = 0.4 P(W) = 0.8 P(M W) = 0.85 a) A married couple watches the show P(M W) = P(M W) P(W) = (0.085)(0.8) = b) A wife watched the show given that her husband does P(M W) P(W M) = = P(M) 0.4 = 0.17 c) At least 1 person of a married couple will watch the show P(M W) = P(M) + P(W) P(M W) = = Problem Set C: Normal Distribution 1. Given a normal distribution with μ = 40 and σ 2 = 0 σ =. Find a. P(X < 32) P(X < 32) = P (z < ) = P (z < 8 ) = P(z < 0.8) = b. P(X > 27) P(X > 27) = 1 P (z < ) = 1 P(z < 1.3) = = c. P(42 < X < 51) P(42 < x < 51) = P(z < 51) P(z < 42) = P (z < ) P (z < = P (z < 11 ) -P (z < 2 ) = P(z < 1.1)-P(z < 0.2) = = ) Prepared by Dr. Francis Joseph H. Campeña 5
6 2. Given a normal distribution with μ = 200 and σ 2 = 0 σ =. Find a. P(X < 214) P(X < 214) = P (z < ) = P(z < 1.4) = b. P(X > 179) P(X > 179) = 1 P(X < 179) = 1 P (z < ) = 1 P(z < 2.1) = = c. P(188 < X < 206) P(188 < X < 206) = P(X < 206) P(X < 188) = P (z < ) P (z < = P(z < 0.6)-P(z < -1.2) ) 3. Adult female have forearm length that are normally distributed with mean 17.5 in and standard deviation of 0.75 in. Remark: Let X be a random variable for the forearm length of an adult female. a. Find the probability that a female s forearm length is between 16 in and 18 in. P(16 < x < 18) = P(x < 18) P(x < 16) = P (z < ) P (x < 0.75 = P(z < 0.67)-P(x < -2.0) = = b. Find the that a female s forearm length is greater than 18.5 P(x > 18.5) = 1 P(x < 18.5) = 1 P (z < ) 0.75 = 1-P(z < 1.33) = = ) A chemical process requires a ph between 5.85 and 7.4. The ph of the process is a normal random variable with mean 6.0 and standard deviation 0.9. Assume that the process must be shut down if the ph will fall outside the acceptable range of 5.85 to Prepared by Dr. Francis Joseph H. Campeña 6
7 Let X be a random variable that denotes the ph level of the chemical process. μ = 6.0 and σ = 0.9 a. What is the probability that the process will not be shut down? Since the threshold of the ph level is from We want to know the probability that the ph level is within this values. P(5.85 < x < 7.40) = P(x < 7.40) P(x < 5.85) = P (z < ) P (x < ) = P(z < 1.56) P(x < 0.17) = = b. What is the probability that the process will be shut down? We let A be the event that the ph level of the process is between 5.58 and 7.40 then A c denotes that the ph level is beyond the threshold of the chemical process and from 4a) P(A) = If we want to get the probability that the process will be shut down we have to get the probability of A c. Thus, P(A c ) = 1 P(A) = = Heights of adult males are normally distributed with mean 65 in and standard deviation of 2.5 in. A male is selected at random. Let X be a random variable that denotes the heights of adult males. μ = 65 and σ = 2.5 a. Find the probability that a male selected have a height greater than 67. P(x > 67) = 1 P(x < 67) = 1 P (z < 2.5 ) = 1 P(z < 0.8) = = b. Find the probability that a male selected have a height less than P(x < 60) = P (z < 2.5 ) = P(z < 2) = Assume that the age at onset of disease X is normally distributed with mean of 50 years and a standard deviation of 15 years. What is the probability that an individual is afflicted with disease X developed it before age 35? Prepared by Dr. Francis Joseph H. Campeña 7
8 Let X be a random variable that denotes the age when an individual is afflicted with the disease X The following data are given: μ = 50 and σ = P(x < 35) = P (z < ) = P(z < 1) = If adult male cholesterol is normally distributed with μ = 200 and σ = 30, what is the probability of selecting a male whose cholesterol is Let X be a random variable that denotes the cholesterol level of an adult male The following data are given: μ = 200 and σ = 30 a. Less than P(x < 165) = P (z < ) 30 = P(z < 1.17) = 0.12 b. Greater than 165 P(x > 165) = 1 P(x < 165) = = c. Between 165 and 220 P(165 < x < 220) = P(x < 220) P(x < 165) = P (z < ) P (z < 30 = P(z < 0.67) P(z < 1.17) = = d. Greater than 220 P(x > 220) = 1 P(x < 220) = 1 P (z < ) 30 = 1 P(z < 0.67) = = ) 30 Prepared by Dr. Francis Joseph H. Campeña 8
Chapter 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 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 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 informationMath 140 Introductory Statistics
5. Models of Random Behavior Math 40 Introductory Statistics Professor Silvia Fernández Chapter 5 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Outcome: Result or answer
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics Professor Silvia Fernández Lecture 8 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. 5.1 Models of Random Behavior Outcome: Result or answer
More informationProbability and Statisitcs
Probability and Statistics Random Variables De La Salle University Francis Joseph Campena, Ph.D. January 25, 2017 Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, 2017 1 / 17 Outline
More informationThe possible experimental outcomes: 1, 2, 3, 4, 5, 6 (Experimental outcomes are also known as sample points)
Chapter 4 Introduction to Probability 1 4.1 Experiments, Counting Rules and Assigning Probabilities Example Rolling a dice you can get the values: S = {1, 2, 3, 4, 5, 6} S is called the sample space. Experiment:
More informationLet us think of the situation as having a 50 sided fair die; any one number is equally likely to appear.
Probability_Homework Answers. Let the sample space consist of the integers through. {, 2, 3,, }. Consider the following events from that Sample Space. Event A: {a number is a multiple of 5 5, 0, 5,, }
More informationLecture 3: Probability
Lecture 3: Probability 28th of October 2015 Lecture 3: Probability 28th of October 2015 1 / 36 Summary of previous lecture Define chance experiment, sample space and event Introduce the concept of the
More informationGeometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last
Geometric Distribution The characteristics of a geometric experiment are: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating
More information8. MORE PROBABILITY; INDEPENDENCE
8. MORE PROBABILITY; INDEPENDENCE Combining Events: The union A B is the event consisting of all outcomes in A or in B or in both. The intersection A B is the event consisting of all outcomes in both A
More informationTopic 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 informationName: Exam 2 Solutions. March 13, 2017
Department of Mathematics University of Notre Dame Math 00 Finite Math Spring 07 Name: Instructors: Conant/Galvin Exam Solutions March, 07 This exam is in two parts on pages and contains problems worth
More informationIntroduction to Probability 2017/18 Supplementary Problems
Introduction to Probability 2017/18 Supplementary Problems Problem 1: Let A and B denote two events with P(A B) 0. Show that P(A) 0 and P(B) 0. A A B implies P(A) P(A B) 0, hence P(A) 0. Similarly B A
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 informationUnit 2 Maths Methods (CAS) Exam
Name: Teacher: Unit Maths Methods (CAS) Exam 1 014 Monday November 17 (9.00-10.45am) Reading time: 15 Minutes Writing time: 90 Minutes Instruction to candidates: Students are permitted to bring into the
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 informationMATH 227 CP 7 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH 227 CP 7 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the mean, µ, for the binomial distribution which has the stated values of n and p.
More informationChapter 2 Solutions Page 12 of 28
Chapter 2 Solutions Page 12 of 28 2.34 Yes, a stem-and-leaf plot provides sufficient information to determine whether a dataset contains an outlier. Because all individual values are shown, it is possible
More information3.2 Probability Rules
3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need
More informationWhat is the probability of getting a heads when flipping a coin
Chapter 2 Probability Probability theory is a branch of mathematics dealing with chance phenomena. The origins of the subject date back to the Italian mathematician Cardano about 1550, and French mathematicians
More informationExercises in Probability Theory Paul Jung MA 485/585-1C Fall 2015 based on material of Nikolai Chernov
Exercises in Probability Theory Paul Jung MA 485/585-1C Fall 2015 based on material of Nikolai Chernov Many of the exercises are taken from two books: R. Durrett, The Essentials of Probability, Duxbury
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 informationBASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES
BASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES COMMON TERMS RELATED TO PROBABILITY Probability is the measure of the likelihood that an event will occur Probability values are
More information1 of 14 7/15/2009 9:25 PM Virtual Laboratories > 2. Probability Spaces > 1 2 3 4 5 6 7 5. Independence As usual, suppose that we have a random experiment with sample space S and probability measure P.
More informationAnnouncements. Topics: To Do:
Announcements Topics: In the Probability and Statistics module: - Sections 1 + 2: Introduction to Stochastic Models - Section 3: Basics of Probability Theory - Section 4: Conditional Probability; Law of
More informationThe enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}
Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment
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 informationChapter 5, 6 and 7: Review Questions: STAT/MATH Consider the experiment of rolling a fair die twice. Find the indicated probabilities.
Chapter5 Chapter 5, 6 and 7: Review Questions: STAT/MATH3379 1. Consider the experiment of rolling a fair die twice. Find the indicated probabilities. (a) One of the dice is a 4. (b) Sum of the dice equals
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 informationPermutation. Permutation. Permutation. Permutation. Permutation
Conditional Probability Consider the possible arrangements of the letters A, B, and C. The possible arrangements are: ABC, ACB, BAC, BCA, CAB, CBA. If the order of the arrangement is important then we
More informationCH 3 P1. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers?
CH 3 P1. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers? P7. The king comes from a family of 2 children. what is
More informationECON Semester 1 PASS Mock Mid-Semester Exam ANSWERS
ECON1310 2006 Semester 1 PASS Mock Mid-Semester Exam ANSWERS MULTIPLE CHOICE QUESTIONS 1. Unemployment rates are an example of: a. Cross-sectional, quantitative, continuous data b. Time-series, quantitative,
More informationTOPIC 12 PROBABILITY SCHEMATIC DIAGRAM
TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationMath/Stat 394 Homework 5
Math/Stat 394 Homework 5 1. If we select two black balls then X 4. This happens with probability ( 4 2). If we select two white balls then X 2. This happens with ( 14 probability (8 2). If we select two
More informationIntro to Probability
Intro to Probability Al Nosedal. University of Toronto. Summer 2017 Al Nosedal. University of Toronto. Intro to Probability Summer 2017 1 / 56 My momma always said: Life was like a box of chocolates. You
More informationProbability Theory: Homework problems
June 22, 2018 Homework 1. Probability Theory: Homework problems 1. A traditional three-digit telephone area code is constructed as follows. The first digit is from the set {2, 3, 4, 5, 6, 7, 8, 9}, the
More informationList the elementary outcomes in each of the following events: EF, E F, F G, EF c, EF G. For this problem, would you care whether the dice are fair?
August 23, 2013 Homework 1. Probability Theory: Homework problems 1. A traditional three-digit telephone area code is constructed as follows. The first digit is from the set 2, 3, 4, 5, 6, 7, 8, 9}, the
More information2.3 Conditional Probability
Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan 2.3 Conditional Probability In this section we introduce the concept of conditional probability. So far, the notation P (A)
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 informationQuantitative Methods for Decision Making
January 14, 2012 Lecture 3 Probability Theory Definition Mutually exclusive events: Two events A and B are mutually exclusive if A B = φ Definition Special Addition Rule: Let A and B be two mutually exclusive
More informationStable matching. Carlos Hurtado. July 5th, Department of Economics University of Illinois at Urbana-Champaign
Stable matching Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu July 5th, 2017 C. Hurtado (UIUC - Economics) Game Theory On the Agenda 1 Introduction
More informationLecture Stat 302 Introduction to Probability - Slides 5
Lecture Stat 302 Introduction to Probability - Slides 5 AD Jan. 2010 AD () Jan. 2010 1 / 20 Conditional Probabilities Conditional Probability. Consider an experiment with sample space S. Let E and F be
More informationHomework 1. Spring 2019 (Due Tuesday January 22)
ECE 302: Probabilistic Methods in Electrical and Computer Engineering Spring 2019 Instructor: Prof. A. R. Reibman Homework 1 Spring 2019 (Due Tuesday January 22) Homework is due on Tuesday January 22 at
More informationTopic -2. Probability. Larson & Farber, Elementary Statistics: Picturing the World, 3e 1
Topic -2 Probability Larson & Farber, Elementary Statistics: Picturing the World, 3e 1 Probability Experiments Experiment : An experiment is an act that can be repeated under given condition. Rolling a
More information4. Conditional Probability
1 of 13 7/15/2009 9:25 PM Virtual Laboratories > 2. Probability Spaces > 1 2 3 4 5 6 7 4. Conditional Probability Definitions and Interpretations The Basic Definition As usual, we start with a random experiment
More informationDiscrete Probability. Chemistry & Physics. Medicine
Discrete Probability The existence of gambling for many centuries is evidence of long-running interest in probability. But a good understanding of probability transcends mere gambling. The mathematics
More informationThe Central Limit Theorem
- The Central Limit Theorem Definition Sampling Distribution of the Mean the probability distribution of sample means, with all samples having the same sample size n. (In general, the sampling distribution
More information2.4. Conditional Probability
2.4. Conditional Probability Objectives. Definition of conditional probability and multiplication rule Total probability Bayes Theorem Example 2.4.1. (#46 p.80 textbook) Suppose an individual is randomly
More informationStatistical Theory 1
Statistical Theory 1 Set Theory and Probability Paolo Bautista September 12, 2017 Set Theory We start by defining terms in Set Theory which will be used in the following sections. Definition 1 A set is
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 information2.6 Tools for Counting sample points
2.6 Tools for Counting sample points When the number of simple events in S is too large, manual enumeration of every sample point in S is tedious or even impossible. (Example) If S contains N equiprobable
More informationChapter 6. Probability
Chapter 6 robability Suppose two six-sided die is rolled and they both land on sixes. Or a coin is flipped and it lands on heads. Or record the color of the next 20 cars to pass an intersection. These
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and sample point
More informationThe set of all outcomes or sample points is called the SAMPLE SPACE of the experiment.
Chapter 7 Probability 7.1 xperiments, Sample Spaces and vents Start with some definitions we will need in our study of probability. An XPRIMN is an activity with an observable result. ossing coins, rolling
More informationTopic 3: Introduction to Probability
Topic 3: Introduction to Probability 1 Contents 1. Introduction 2. Simple Definitions 3. Types of Probability 4. Theorems of Probability 5. Probabilities under conditions of statistically independent events
More informationMath Treibergs. Solutions to Fourth Homework February 13, 2009
Math 5010 1. Treibergs Solutions to Fourth Homework February 13, 2009 76[12] Two roads join Ayton to Beaton, and two further roads join Beaton to the City. Ayton is directly connected to the City by railway.
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 information( ) P A B : Probability of A given B. Probability that A happens
A B A or B One or the other or both occurs At least one of A or B occurs Probability Review A B A and B Both A and B occur ( ) P A B : Probability of A given B. Probability that A happens given that B
More informationBasic Statistics and Probability Chapter 3: Probability
Basic Statistics and Probability Chapter 3: Probability Events, Sample Spaces and Probability Unions and Intersections Complementary Events Additive Rule. Mutually Exclusive Events Conditional Probability
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 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 informationProbability Problems for Group 3(Due by EOC Mar. 6)
Probability Problems for Group (Due by EO Mar. 6) Bob And arol And Ted And Alice And The Saga ontinues. Six married couples are standing in a room. a) If two people are chosen at random, find the probability
More informationMATH 3C: MIDTERM 1 REVIEW. 1. Counting
MATH 3C: MIDTERM REVIEW JOE HUGHES. Counting. Imagine that a sports betting pool is run in the following way: there are 20 teams, 2 weeks, and each week you pick a team to win. However, you can t pick
More informationTwo-Sided Matching. Terence Johnson. December 1, University of Notre Dame. Terence Johnson (ND) Two-Sided Matching December 1, / 47
Two-Sided Matching Terence Johnson University of Notre Dame December 1, 2017 Terence Johnson (ND) Two-Sided Matching December 1, 2017 1 / 47 Markets without money What do you do when you can t use money
More informationProbability, Conditional Probability and Bayes Rule IE231 - Lecture Notes 3 Mar 6, 2018
Probability, Conditional Probability and Bayes Rule IE31 - Lecture Notes 3 Mar 6, 018 #Introduction Let s recall some probability concepts. Probability is the quantification of uncertainty. For instance
More information2. Conditional Probability
ENGG 2430 / ESTR 2004: Probability and Statistics Spring 2019 2. Conditional Probability Andrej Bogdanov Coins game Toss 3 coins. You win if at least two come out heads. S = { HHH, HHT, HTH, HTT, THH,
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 informationMath 2311 Test 1 Review. 1. State whether each situation is categorical or quantitative. If quantitative, state whether it s discrete or continuous.
Math 2311 Test 1 Review Know all definitions! 1. State whether each situation is categorical or quantitative. If quantitative, state whether it s discrete or continuous. a. The amount a person grew (in
More informationProbability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary)
Chapter 14 From Randomness to Probability How to measure a likelihood of an event? How likely is it to answer correctly one out of two true-false questions on a quiz? Is it more, less, or equally likely
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 informationDiscrete Probability Models
CHAPTER ONE Discrete Probability Models 1.1 INTRODUCTION The mathematical study of probability can be traced to the seventeenth-century correspondence between Blaise Pascal and Pierre de Fermat, French
More informationExample. What is the sample space for flipping a fair coin? Rolling a 6-sided die? Find the event E where E = {x x has exactly one head}
Chapter 7 Notes 1 (c) Epstein, 2013 CHAPTER 7: PROBABILITY 7.1: Experiments, Sample Spaces and Events Chapter 7 Notes 2 (c) Epstein, 2013 What is the sample space for flipping a fair coin three times?
More informationIntermediate Math Circles November 8, 2017 Probability II
Intersection of Events and Independence Consider two groups of pairs of events Intermediate Math Circles November 8, 017 Probability II Group 1 (Dependent Events) A = {a sales associate has training} B
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 informationMore on conditioning and Mr. Bayes
More on conditioning and Mr. Bayes Saad Mneimneh 1 Multiplication rule for conditioning We can generalize the formula P(A,B) P(A B)P(B) to more than two events. For instance, P(A,B,C) P(A)P(B A)P(C A,B).
More information(i) Given that a student is female, what is the probability of having a GPA of at least 3.0?
MATH 382 Conditional Probability Dr. Neal, WKU We now shall consider probabilities of events that are restricted within a subset that is smaller than the entire sample space Ω. For example, let Ω be the
More information4.4-Multiplication Rule: Basics
.-Multiplication Rule: Basics The basic multiplication rule is used for finding P (A and, that is, the probability that event A occurs in a first trial and event B occurs in a second trial. If the outcome
More information(6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6)
Section 7.3: Compound Events Because we are using the framework of set theory to analyze probability, we can use unions, intersections and complements to break complex events into compositions of events
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 informationConditional Probability. CS231 Dianna Xu
Conditional Probability CS231 Dianna Xu 1 Boy or Girl? A couple has two children, one of them is a girl. What is the probability that the other one is also a girl? Assuming 50/50 chances of conceiving
More informationMath 710 Homework 1. Austin Mohr September 2, 2010
Math 710 Homework 1 Austin Mohr September 2, 2010 1 For the following random experiments, describe the sample space Ω For each experiment, describe also two subsets (events) that might be of interest,
More informationStat 225 Week 2, 8/27/12-8/31/12, Notes: Independence and Bayes Rule
Stat 225 Week 2, 8/27/12-8/31/12, Notes: Independence and Bayes Rule The Fall 2012 Stat 225 T.A.s September 7, 2012 1 Monday, 8/27/12, Notes on Independence In general, a conditional probability will change
More informationProbability and Sample space
Probability and Sample space We call a phenomenon random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. The probability of any outcome
More informationSTAT:5100 (22S:193) Statistical Inference I
STAT:5100 (22S:193) Statistical Inference I Week 3 Luke Tierney University of Iowa Fall 2015 Luke Tierney (U Iowa) STAT:5100 (22S:193) Statistical Inference I Fall 2015 1 Recap Matching problem Generalized
More information1 Probability Theory. 1.1 Introduction
1 Probability Theory Probability theory is used as a tool in statistics. It helps to evaluate the reliability of our conclusions about the population when we have only information about a sample. Probability
More informationFrancine s bone density is 1.45 standard deviations below the mean hip bone density for 25-year-old women of 956 grams/cm 2.
Chapter 3 Solutions 3.1 3.2 3.3 87% of the girls her daughter s age weigh the same or less than she does and 67% of girls her daughter s age are her height or shorter. According to the Los Angeles Times,
More informationConditional Probability Solutions STAT-UB.0103 Statistics for Business Control and Regression Models
Conditional Probability Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Counting (Review) 1. There are 10 people in a club. How many ways are there to choose the following:
More informationMath 227 Test 2 Ch5. Name
Math 227 Test 2 Ch5 Name Find the mean of the given probability distribution. 1) In a certain town, 30% of adults have a college degree. The accompanying table describes the probability distribution for
More informationUNIT 5 ~ Probability: What Are the Chances? 1
UNIT 5 ~ Probability: What Are the Chances? 1 6.1: Simulation Simulation: The of chance behavior, based on a that accurately reflects the phenomenon under consideration. (ex 1) Suppose we are interested
More informationMultiple Choice Practice Set 1
Multiple Choice Practice Set 1 This set of questions covers material from Chapter 1. Multiple choice is the same format as for the midterm. Q1. Two events each have probability 0.2 of occurring and are
More informationMath 461 B/C, Spring 2009 Midterm Exam 1 Solutions and Comments
Math 461 B/C, Spring 2009 Midterm Exam 1 Solutions and Comments 1. Suppose A, B and C are events with P (A) = P (B) = P (C) = 1/3, P (AB) = P (AC) = P (BC) = 1/4 and P (ABC) = 1/5. For each of the following
More informationTest 3 SOLUTIONS. x P(x) xp(x)
16 1. A couple of weeks ago in class, each of you took three quizzes where you randomly guessed the answers to each question. There were eight questions on each quiz, and four possible answers to each
More informationQUIZ 1 (CHAPTERS 1-4) SOLUTIONS MATH 119 FALL 2012 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS
QUIZ 1 (CHAPTERS 1-4) SOLUTIONS MATH 119 FALL 2012 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% Show all work, simplify as appropriate, and use good form and procedure (as in class). Box in your final
More informationSingle Maths B: Introduction to Probability
Single Maths B: Introduction to Probability Overview Lecturer Email Office Homework Webpage Dr Jonathan Cumming j.a.cumming@durham.ac.uk CM233 None! http://maths.dur.ac.uk/stats/people/jac/singleb/ 1 Introduction
More informationEvent A: at least one tail observed A:
Chapter 3 Probability 3.1 Events, sample space, and probability Basic definitions: An is an act of observation that leads to a single outcome that cannot be predicted with certainty. A (or simple event)
More informationWhat does a population that is normally distributed look like? = 80 and = 10
What does a population that is normally distributed look like? = 80 and = 10 50 60 70 80 90 100 110 X Empirical Rule 68% 95% 99.7% 68-95-99.7% RULE Empirical Rule restated 68% of the data values fall within
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 informationOutline Conditional Probability The Law of Total Probability and Bayes Theorem Independent Events. Week 4 Classical Probability, Part II
Week 4 Classical Probability, Part II Week 4 Objectives This week we continue covering topics from classical probability. The notion of conditional probability is presented first. Important results/tools
More information