frequency of E sample size


 Emory Gilmore
 2 years ago
 Views:
Transcription
1 Chapter 4 Probability (Page 1 of 24) 4.1 What is Probability? Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. Probabilities closer to 1 indicate that the event is more likely to occur. Probabilities near 0 indicate that the event is less likely to occur. Probability Notation P( A), read P of A, denotes the probability of event A. P( A) = 1 means event A is certain to occur P( A) = 0 means event A is impossible. Probability based on Relative Frequency The probability of an event E occurring can be estimated by the relative frequency of the occurrence of the event. P(E) = frequency of E sample size = f n Example A From a random sample of 100 lab reports 40 had erroneous results. What is the probability that a lab report selected at random has an erroneous result? If event E is has an erroneous result, then P(E) = f n = = 0.40 So, approximately 40% of lab reports are erroneous. The estimate gets better as n gets larger and larger.
2 Chapter 4 Probability (Page 2 of 24) Law of Large Numbers In the long run, as the sample size increases, the relative frequency of outcomes gets closer and closer to the actual probability. Example B 1. Estimate the probability of heads occurring when tossing a coin. Simulate the experiment by using the TI83 random number generator. STAT / 2: SortA( MATH / PRB / 5: randint(min.range, max.range, # of numbers) a. Use 5 tosses b. Use 25 tosses c. Use 200 tosses Probability when the Outcomes are Equally Likely Probability of event E = Number of outcomes favorable to E Total number of outcomes Example C a. What is the probability of rolling a 3 on a die? b. What is the probability of rolling a 2 or 4 on a die?
3 Chapter 4 Probability (Page 3 of 24) Guided Exercise 1 Assign a probability for the following events and state if the technique was intuition, relative frequency, or the formula for equally likely outcomes. a. The health center director at a college found that 375 students out of 500 tested needed corrective lenses. What is the probability that a randomly selected student from the college needs corrective lenses? b. Four members on a cleanup committee draw lots to see who will clean the barbecue grills. What is the probability that George will clean the grill? c. Brad Gilbert, a tennis commentator, said that Roger Federer had an advantage over Marcos Baghdatis in winning the 2006 Australian Open. What do you suppose is a reasonable number for the probability that Federer wins (before the outcome is known).
4 Chapter 4 Probability (Page 4 of 24) Statistical Experiment Terminology a. A statistical experiment or statistical observation is any random activity that results in a definite outcome. b. An outcome (simple event) is the observable or measurable result of an experiment. c. The sample space, S, is the set of all possible outcomes of an experiment. d. An event E is any set of outcomes. e. P(E) = n(e) Number of Outcomes Favorable to E n(s) Total Number of Outcomes Example D Identify the experiment, sample space and event. Then answer the question. 1. A coin is tossed, what is the probability of it landing heads up? 2. A die is tossed, what is the probability of a 2 or 4? 3. A card is drawn from a standard deck of 52 cards, what is the probability that a diamond is drawn. 4. Two dice are tossed, what is the probability that the sum is 6?
5 Chapter 4 Probability (Page 5 of 24) Example 1 Eye color is controlled by a pair of genes (one from the mother and one from the father) called a genotype. Brown eye color, B, is dominant over blue eye color, b. Therefore, in the genotype Bb the browneye gene, B, dominates and a person with a Bb genotype has brown eyes. If both parents have brown eyes and genotype Bb, what is the probability that the child will have blue eyes? What is the probability that the child will have brown eyes? Guided Exercise 2 Professor Gill has 3 truefalse questions on an exam. In order to assure the pattern of tf answers is random he lists all combinations of three tf answers on sheets of paper and randomly selects one from a hat. a. List the sample space in a tree diagram. b. What is P(all three answers will be false)? c. What is P(exactly two will be true)?
6 Chapter 4 Probability (Page 6 of 24) Complement of an Event If P( A) is the probability that event A will occur, then the complement of that, P( A c ), is the probability that event A will not occur. So, A and A c together make up the sample space, and 1. P( A c ) = 1! P( A) 2. P( A) + P( A c ) = 1 Guided Exercise 3 If you breed two creamcolored guinea pigs, the probability that the offspring will be pure white is What is the probability that the offspring will not be pure white?
7 Chapter 4 Probability (Page 7 of 24) Probability Summary 1. The probability of event A is denoted P( A). 2. 0! P( A)! 1 3. The sum of the probabilities of all outcomes in a sample space is P( A) + P( A c ) = 1 5. If P( A) = 0, then event A is impossible. 6. If P( A) = 1, then event A is certain. Probability versus Statistics Probability is the field through which statistical work is done. * Probability is the field of study that makes statements about what will occur when samples are drawn from a known population. * Statistics is the field of study that describes how samples are to be obtained and how inferences are to be made about unknown populations. Example of Probability Questions Suppose a box contains 3 green balls, 5 red balls, and 4 white balls. a. If one ball is drawn, what is P(green)? b. If 3 balls are drawn, what is P(one is white and two are red)? c. If 4 balls are drawn, what is P(none are red)? Example of a Statistics Exploration Suppose a box contains a collection of colored balls. The central question is to discover the (relative) number and color of each ball in the box by random sampling.
8 Chapter 4 Probability Rules (Page 8 of 24) 4.2 Compound Events (2 or more events) Independent Events Two events are independent if the outcome of one event does not affect the outcome of the other event (and Visa Versa). Mathematically, two events A and B are independent if and only if P(A and B)= P(A)! P(B) Example E Identify if the events described are independent or not. a. Roll a fair die twice. b. Draw two cards from a standard deck of 52 cards, without replacing the first card before drawing the second. Multiplication Rules: Probability of Event A and Event B 1. Two Events A and B are independent if and only if P(A and B)= P(A)! P(B)) 2. For Any Events A and B [dependent or independent] P(A and B)= P(A)! P(B, given A has ocurred)= P(A)! P(B A) Example 3 Suppose two fair dice are thrown. What is the probability of getting a five on each die? a. Solve using the multiplication rule. b. Solve by drawing the sample space.
9 Chapter 4 Probability Rules (Page 9 of 24) Example 4 Find the probability of drawing two Aces from a standard deck of 52 cards if a. the first card is replaced into the deck and the deck is shuffled before drawing the second card (called with replacement ). b. the cards are drawn in sequence without replacement of the first card (called without replacement ). c. In part b, explain why (mathematically) the two events are not independent. Guided Exercise 4 Andrew is 55, and the probability that he will be alive in 10 years is Ellen is 35, and the probability that she will be alive in 10 years is What is the probability that both will be alive in 10 years?
10 Chapter 4 Probability Rules (Page 10 of 24) Guided Exercise 5 A quality control procedure for testing ReadyFlash disposable cameras is done by randomly drawing 2 cameras from each lot of 100 (w/o replacement). If both are defective, then the entire lot is rejected. Find the probability that the lot will be rejected (i.e. both cameras will be defective) if the lot contains 10 defective cameras. a. What is the probability of getting a defective camera on the first draw? b. What is the probability of getting a defective camera on the second draw? c. Find the probability that the lot will be rejected (i.e. both cameras will be defective) if the lot contains 10 defective cameras. Example D Suppose a fair coin is tossed, then a fair die is rolled, and finally a card is drawn from a standard deck of 52 cards. What is the probability of the outcome of heads on the coins and 5 on the die and an ace for the card?
11 Chapter 4 Probability Rules (Page 11 of 24) Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time. Mutually Exclusive Events Events that can occur together A B A B Sample Space, S Sample Space, S Example E Suppose a single card is drawn from a wellshuffled bridge deck. a. Are the events of drawing a Jack or King mutually exclusive? b. Are the events of drawing a Jack or Diamond mutually exclusive? Addition Rules: Probability of Event A or Event B 1. For mutually exclusive events A and B P(A or B)= P(A)+ P(B) 2. For any events A and B P(A or B)= P(A)+ P(B) P(A and B)
12 Chapter 4 Probability Rules (Page 12 of 24) Addition Rules: Probability of Event A or Event B 1. For mutually exclusive events A and B P(A or B)= P(A)+ P(B) 2. For any events A and B P(A or B)= P(A)+ P(B) P(A and B) Example F a. What is the probability of drawing a Jack or a King from a standard deck in one draw? b. What is the probability of drawing a Jack or a diamond from a standard deck in one draw? Guided Exercise 7 If you purchase a pair of slacks in your waist size from a second hand clothing store the probability that they will be too tight is 0.30 and the probability that they will be too loose is a. Are the events mutually exclusive? b. If you choose a pair of slacks in your waist size, what is the probability that they will be too tight or too loose?
13 Chapter 4 Probability Rules (Page 13 of 24) Guided Exercise 8 In a program to prepare for a high school equivalency exam it is found that 80% of the students need work in math, 70% in English, and 55% in both areas. Draw a Venn diagram and find the probability that a randomly selected student will need work in a. Math and English b. Math or English c. Math, but not English d. English, but not Math e. Neither Math nor English Exercise 14 About 14% of senior citizens (65 years or older) get the flu each year, and about 24% of the people under 65 years old get the flu each year. In the general population, there are 12.5% senior citizens. What is the probability that a person selected at random a. is a senior citizen who will get the flu? b. is a person under 65 who will get the flu? c. Draw a tree diagram for this problem.
14 Chapter 4 Probability Rules (Page 14 of 24) Exercise 20 Diagnostic tests of medical conditions have several results. The test result can be positive of negative, whether or not the patient has the condition (+ indicates the patient tested positive for the condition). Consider a random sample of 200 patients, some of whom have a medical condition and some of whom do not. Results of a new diagnostic test for the condition are shown. For a person selected at random compute the following. a. P(+, given condition present) b. P(, given condition present) c. P(, given condition absent) d. P(+, given condition absent) e. P(condition present and +) f. P(condition present and ) Condition Present Condition Absent Row Total Test Test Column Total g. Are the events Test + and Condition Absent independent? Explain mathematically.
15 Chapter 4 Probability Rules (Page 15 of 24) Exercise 26 An alcoholic treatment program has two phases: Phase 1 lasts 10 weeks; phase 2 lasts 1 year. The probability that a client will relapse in phase 1 is However, if the client did NOT have a relapse in phase 1, then the probability that a client will have a relapse in phase 2 is If the client did have a relapse in phase 1, then the probability that a client will have a relapse in phase 2 is Let R 1 be the event that the client had a relapse in phase 1, and R 2 be the event that the client had a relapse in phase 2. Draw a tree diagram showing all possible outcomes, and find the probability of each of the following. a. P(R 1 ), P(not R 1 ) b. P(R 2, given not R 1 ), P(not R 2, given not R 1 ) c. P(R 2, given not R 1 ), P(not R 2, given not R 1 ) e. P(R 2 )
16 Chapter 4 Trees & Counting Techniques (Page 16 of 24) 4.3 Tree Diagrams and Counting Techniques Example 7 Jackie needs to take psychology, anatomy, and Spanish. There are 4 sections of psychology, 2 of anatomy, and 3 of Spanish offered that do not conflict with each other. Use a tree diagram to list all the possible schedules Jackie can take. How many possible schedules can she take? Guided Exercise 10 Louis plays three tennis matches. Use a tree diagram to list all the possible outcomes. How many possible outcomes can occur?
17 Chapter 4 Trees & Counting Techniques (Page 17 of 24) Example 8 a. Suppose there are 5 balls of identical size in an urn: 3 red and 2 blue. You are asked to draw out one ball, note its color and set it aside. Then draw another ball and note its color. List all possible outcomes and the probability for each outcome. b. Repeat, except replace the first ball before drawing the second ball.
18 Chapter 4 Trees & Counting Techniques (Page 18 of 24) Multiplication Rule of Counting If there are m possible outcomes for event E 1 and n possible outcomes for event E 2, then there are a total of m!n outcomes for events E 1 followed by E 2. Example 9 An automobile comes in a choice of two body styles, 3 interior packages, 4 colors, and 2 types of transmission. How many distinct car orders are possible? Guided Exercise 12 A menu has a choice of 2 appetizers, 3 main courses, and 4 desserts. How many different full meals can be ordered?
19 Chapter 4 Trees & Counting Techniques (Page 19 of 24) Example G How many different ways can 4 people be seated at a dinner table with 4 chairs? Factorials For positive integer n, n!= 1!2!3!!(n "1)! n 0!= 1 e.g. 1!= 1 2!= 1!2 = 2 3!= 1!2!3 = 6 4!= 1!2!3!4 = 24 Example H Compute a. 6! b. 3!4! c. 8! (8! 5)! d. 10! 2!(10! 2)!
20 Chapter 4 Trees & Counting Techniques (Page 20 of 24) Permutations Ordered Arrangements A permutation is an ordered arrangement. The number of ways to arrange n distinct objects, taken r at a time, is n P r = P n,r = n! (n! r)! (MATH>PRB>2: npr) Example I How many different ways can 4 people be seated at a dinner table with 4 chairs? Example J How many different ways can 6 people be seated at a dinner table with 6 chairs? Example K How many different ways can 8 people be seated at a dinner table with 8 chairs?
21 Chapter 4 Trees & Counting Techniques (Page 21 of 24) Example L Alan, Bob, Cathy and Diane are friends. Find the number of ways to arrange a. all 4 in a line for a picture. List all the permutations. ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA b. 3 of 4 in a line for a picture. List all the permutations. c. 2 of 4 in a line for a picture. List all the permutations. d. 1 of 4 in a line for a picture. List all the permutations. e. 0 of 4 in a line for a picture. List all the permutations.
22 Chapter 4 Trees & Counting Techniques (Page 22 of 24) Example 10 Find the number of ways to arrange 8 objects taken {8, 7, 6, 5, 4, 3, 2, 1, and 0} at a time. Guided Exercise 14 A board of directors has 12 members and must elect a president, vice president, and treasurer. In how many ways can this be done if no person can hold more than one office? Example M A committee has 5 members. In how many ways can a subcommittee of size 3 be selected? Size 2? Size 1? Size 4? Size 5? Size 0?
23 Chapter 4 Trees & Counting Techniques (Page 23 of 24) Combinations Order Does Not Matter A combination is a subset, or subgroup, in which the order does not matter. The number of combinations of n objects taken r at a time is n C r = C n,r = n! r!!(n " r)! (MATH>PRB> 2:nCr) Permutations vs. Combinations 1. Permutations consider groupings and order. 2. Combinations consider groupings only. 3. n P r! n C r. That is, the number of permutations is greater than or equal to the number of combinations. Example I Suppose a committee has 5 members: Alice, Bob, Cathy, Dave, and Eve. How many subcommittees of size 0, 1, 2, 3, 4, and 5 can be formed. List all the possible subcommittees.
24 Chapter 4 Trees & Counting Techniques (Page 24 of 24) Example 11 A board of directors has 12 members and must send three to a convention. In how many ways can this be done? Guided Exercise 15 In how many different ways can you select 4 books from a list of 10 books? Open your books to exercises #2230 in section 4.3. Show your work by showing the proper probability notation and the solution. Circle your answer. On exams I will not say whether to use a permutation or combination identifying that is central to the problem.
Lectures 2 and 3 : Probability
Lectures 2 and 3 : Probability Jonathan L. Marchini October 15, 2004 In this lecture we will learn about why we need to learn about probability what probability is how to assign probabilities how to manipulate
More information15 Chances, Probabilities, and Odds
15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces
More informationChapter 5  Probability
Chapter 5  Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set
More informationChapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.
MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.
More informationThe Mathematics of Game Shows
The Mathematics of Game Shows Bowen Kerins Research Scientist, EDC (and parttime game consultant) bkerins@edc.org Overview Game shows are filled with math problems Contestants How do I play best? How
More informationLesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationProbability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.
1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event
More informationA (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.
Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome
More informationThe study of probability has increased in popularity over the years because of its wide range of practical applications.
6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,
More information33 Probability: Some Basic Terms
33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been
More informationExample: If we roll a dice and flip a coin, how many outcomes are possible?
12.5 Tree Diagrams Sample space Sample point Counting principle Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible
More informationSection 6.2 Definition of Probability
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical
More informationEXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS
EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More information+ Section 6.2 and 6.3
Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities
More informationProbability. Experiment  any happening for which the result is uncertain. Outcome the possible result of the experiment
Probability Definitions: Experiment  any happening for which the result is uncertain Outcome the possible result of the experiment Sample space the set of all possible outcomes of the experiment Event
More informationMost of us would probably believe they are the same, it would not make a difference. But, in fact, they are different. Let s see how.
PROBABILITY If someone told you the odds of an event A occurring are 3 to 5 and the probability of another event B occurring was 3/5, which do you think is a better bet? Most of us would probably believe
More information7.5 Conditional Probability; Independent Events
7.5 Conditional Probability; Independent Events Conditional Probability Example 1. Suppose there are two boxes, A and B containing some red and blue stones. The following table gives the number of stones
More informationBasic Probability Theory I
A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population
More informationnumber of equally likely " desired " outcomes numberof " successes " OR
Math 107 Probability and Experiments Events or Outcomes in a Sample Space: Probability: Notation: P(event occurring) = numberof waystheevent canoccur total number of equally likely outcomes number of equally
More informationSection Tree Diagrams. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 12.5 Tree Diagrams What You Will Learn Counting Principle Tree Diagrams 12.52 Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed
More informationProbability. Vocabulary
MAT 142 College Mathematics Probability Module #PM Terri L. Miller & Elizabeth E. K. Jones revised January 5, 2011 Vocabulary In order to discuss probability we will need a fair bit of vocabulary. Probability
More informationLesson 1: Review of Decimals: Addition, Subtraction, Multiplication
LESSON 1: REVIEW OF DECIMALS: ADDITION AND SUBTRACTION Weekly Focus: whole numbers and decimals Weekly Skill: place value, add, subtract, multiply Lesson Summary: In the warm up, students will solve a
More informationProbability and Venn diagrams UNCORRECTED PAGE PROOFS
Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve
More informationWhat is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts
Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called
More informationSection 42 Basic Concepts of Probability
Section 42 Basic Concepts of Probability 4.11 Events and Sample Space Event any collection of results or outcomes of a procedure Outcome (simple event) It cannot be further broken down into simpler components
More informationhttps://assessment.casa.uh.edu/assessment/printtest.htm PRINTABLE VERSION Quiz 10
1 of 8 4/9/2013 8:17 AM PRINTABLE VERSION Quiz 10 Question 1 Let A and B be events in a sample space S such that P(A) = 0.34, P(B) = 0.39 and P(A B) = 0.19. Find P(A B). a) 0.4872 b) 0.5588 c) 0.0256 d)
More information4.3. Addition and Multiplication Laws of Probability. Introduction. Prerequisites. Learning Outcomes. Learning Style
Addition and Multiplication Laws of Probability 4.3 Introduction When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring
More informationMath 141. Lecture 1: Conditional Probability. Albyn Jones 1. 1 Library jones/courses/141
Math 141 Lecture 1: Conditional Probability Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Definitions: Sample Space, Events Last Time Definitions: Sample Space,
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationTopic 6: Conditional Probability and Independence
Topic 6: September 1520, 2011 One of the most important concepts in the theory of probability is based on the question: How do we modify the probability of an event in light of the fact that something
More informationConditional Probability
EE304  Probability and Statistics October 7, 2010 We are often interested in the likelihood of an event occurring given that another has occurred. s of this type include: Probability that a train arrives
More informationAn event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event
An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationChapter 3. Probability
Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.
More informationPROBABILITY. Chapter Overview
Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability
More informationBasic Probability. Probability: The part of Mathematics devoted to quantify uncertainty
AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationMATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics
MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you should
More informationPROBABILITY 14.3. section. The Probability of an Event
4.3 Probability (43) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques
More informationIn this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events.
Lecture#4 Chapter 4: Probability In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events. 42 Fundamentals Definitions:
More informationMath 421: Probability and Statistics I Note Set 2
Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the
More informationSection 65 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More informationSTAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia
STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that
More informationMath 3C Homework 3 Solutions
Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard
More informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
More informationMATH 3070 Introduction to Probability and Statistics Lecture notes Probability
Objectives: MATH 3070 Introduction to Probability and Statistics Lecture notes Probability 1. Learn the basic concepts of probability 2. Learn the basic vocabulary for probability 3. Identify the sample
More informationMath 210. 1. Compute C(1000,2) (a) 499500. (b) 1000000. (c) 2. (d) 999000. (e) None of the above.
Math 210 1. Compute C(1000,2) (a) 499500. (b) 1000000. (c) 2. (d) 999000. 2. Suppose that 80% of students taking calculus have previously had a trigonometry course. Of those that did, 75% pass their calculus
More informationProbability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes
Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all
More informationP(A) = s n = Definitions. P  denotes a probability. A, B, and C  denote specific events. P (A)  Chapter 4 Probability. Notation for Probabilities
Chapter 4 Probability Slide 1 Definitions Slide 2 41 Overview 42 Fundamentals 43 Addition Rule 44 Multiplication Rule: Basics 45 Multiplication Rule: Complements and Conditional Probability 46 Probabilities
More informationOdds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes.
MATH 11008: Odds and Expected Value Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes. Suppose all outcomes in a sample space are equally likely where a of them
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More information7.1 Sample space, events, probability
7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationAP Stats  Probability Review
AP Stats  Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationMethods Used for Counting
COUNTING METHODS From our preliminary work in probability, we often found ourselves wondering how many different scenarios there were in a given situation. In the beginning of that chapter, we merely tried
More informationMath 117 Chapter 7 Sets and Probability
Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a welldefined collection of specific objects. Each item in the set is called an element or a member. Curly
More informationReview of Probability
Review of Probability Table of Contents Part I: Basic Equations and Notions Sample space Event Mutually exclusive Probability Conditional probability Independence Addition rule Multiplicative rule Using
More informationRemember to leave your answers as unreduced fractions.
Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,
More information34 Probability and Counting Techniques
34 Probability and Counting Techniques If you recall that the classical probability of an event E S is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements of E and S respectively.
More informationA Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability  1. Probability and Counting Rules
Probability and Counting Rules researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people in this random sample
More informationQuestion of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay
QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know
More informationProbability: Events and Probabilities
Probability: Events and Probabilities PROBABILITY: longrun relative frequency; likelihood or chance that an outcome will happen. A probability is a number between 0 and 1, inclusive, EVENT: An outcome
More informationA probability experiment is a chance process that leads to welldefined outcomes. 3) What is the difference between an outcome and an event?
Ch 4.2 pg.191~(110 all), 12 (a, c, e, g), 13, 14, (a, b, c, d, e, h, i, j), 17, 21, 25, 31, 32. 1) What is a probability experiment? A probability experiment is a chance process that leads to welldefined
More informationLesson 1: Experimental and Theoretical Probability
Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.
More informationChapter 5 A Survey of Probability Concepts
Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible
More informationFormula Sheet About Exam Study Tips Hub: Math Teacher Resources Index Old Exams Graphing Calculator Guidelines SED Links Math A Topic Headings
Regents Prep Math A Formula Sheet About Exam Study Tips Hub: Math Teacher Resources Index Old Exams Graphing Calculator Guidelines SED Links Math A Topic Headings Work is underway to create sites for the
More informationStatistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then
Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance
More informationSection 8.1 Properties of Probability
Section 8. Properties of Probability Section 8. Properties of Probability A probability is a function that assigns a value between 0 and to an event, describing the likelihood of that event happening.
More informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More informationPROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA
PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet
More informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
More informationIAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics
More informationChapter 4: Probabilities and Proportions
Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,
More informationIf we know that LeBron s next field goal attempt will be made in a game after 3 days or more rest, it would be natural to use the statistic
Section 7.4: Conditional Probability and Tree Diagrams Sometimes our computation of the probability of an event is changed by the knowledge that a related event has occurred (or is guaranteed to occur)
More information7 Probability. Copyright Cengage Learning. All rights reserved.
7 Probability Copyright Cengage Learning. All rights reserved. 7.1 Sample Spaces and Events Copyright Cengage Learning. All rights reserved. Sample Spaces 3 Sample Spaces At the beginning of a football
More informationCounting principle, permutations, combinations, probabilities
Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing
More informationOptimal Scheduling of Flowshop Batch Process for ZeroWait and No Intermediate Storage Transfer Policy
Optimal Scheduling of Flowshop Batch Process for ZeroWait and No Intermediate Storage Transfer Policy H. Abdel Samad Teaching Assistant Chemical Department High Institute of Shorouk City H. Moselhy Associated
More informationChapter 3: Probability
Chapter 3: Probability We see probabilities almost every day in our real lives. Most times you pick up the newspaper or read the news on the internet, you encounter probability. There is a 65% chance of
More informationProbability. What is the probability that Christ will come again to judge all mankind?
Probability What is the probability that Christ will come again to judge all mankind? Vocabulary Words Definition of Probability English definition Math definition Inequality range for Probability Sample
More information**Chance behavior is in the short run but has a regular and predictable pattern in the long run. This is the basis for the idea of probability.
AP Statistics Chapter 5 Notes 5.1 Randomness, Probability,and Simulation In tennis, a coin toss is used to decide which player will serve first. Many other sports use this method because it seems like
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationChapter 9 Section 1 Probability
Name Guided Notes Chapter 9 Chapter 9 Section 1 Probability Outcome: the of an action Event: a collection of outcomes Theoretical Probability: when all outcomes are likely P(event) = number of favorable
More informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a datagenerating
More informationPROBABILITY. Chapter Overview Conditional Probability
PROBABILITY Chapter. Overview.. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the
More informationMassachusetts Institute of Technology
n (i) m m (ii) n m ( (iii) n n n n (iv) m m Massachusetts Institute of Technology 6.0/6.: Probabilistic Systems Analysis (Quiz Solutions Spring 009) Question Multiple Choice Questions: CLEARLY circle the
More information36 Odds, Expected Value, and Conditional Probability
36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face
More information4.4 Conditional Probability
4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.
More informationUnit 1: Probability. Experimental Probability:  probability that came from a simulation such as tossing dice, coins etc.
pplied Math 0 Unit : Probability Unit : Probability.: Experimental and Theoretical Probability Experimental Probability:  probability that came from a simulation such as tossing dice, coins etc. inomial
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationSample Space, Events, and PROBABILITY
Sample Space, Events, and PROBABILITY In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationContemporary Mathematics MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More informationSECTION 105 Multiplication Principle, Permutations, and Combinations
105 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial
More informationProbability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2
Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability
More information4.5 Finding Probability Using Tree Diagrams and Outcome Tables
4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or
More informationBasic concepts in probability. Sue Gordon
Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are
More informationExample. For example, if we roll a die
3 Probability A random experiment has an unknown outcome, but a well defined set of possible outcomes S. The set S is called the sample set. An element of the sample set S is called a sample point (elementary
More information3. Conditional probability & independence
3. Conditional probability & independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Derivation: Suppose we repeat the experiment n times. Let
More information