Chapter 3: Elements of Chance: Probability Methods

Size: px
Start display at page:

Transcription

1 Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week

2 Introduction In this chapter we will focus on the definitions of random experiment, outcome, and event, probability and its rules, and an important result that has many applications to management decision making:.

3 Definition: A random experiment is a process leading to two or more possible outcomes, without knowing exactly which outcome will occur. Some examples of random experiments: A coin is tossed and the outcome is either a head (H) or a tail (T). A company has the possibility of receiving 0 5 contract awards. A die is rolled and the outcome is one of the six sides of the die. A customer enters a store and either purchases a shirt or does not.

4 Definition: The possible outcomes of a random experiment are called the basic outcomes and the set of all basic outcomes is called the sample space and is denoted by S. In the experiment of tossing a coin the basic outcomes are head (H) and tail (T), so S = {H, T }, rolling a die the basic outcomes are 1, 2, 3, 4, 5, and 6, so S = {1, 2, 3, 4, 5, 6}.

5 Definition: An event, E, is any subset of the sample space consisting of basic outcomes. The null (impossible) event represents the absence of a basic outcome and is denoted by. An event is said to occur if the random experiment results in one of the basic outcomes in that event. In the experiment of rolling a die the basic outcomes are 1, 2, 3, 4, 5, and 6, so S = {1, 2, 3, 4, 5, 6}. Let A be the event that the number on the die is even and B be the event that the number on the die is odd. Then A = {2, 4, 6} S and B = {1, 3, 5} S. Event A is said to be occur if and only if the result of the experiment is one of 2, 4, or 6. Similarly, event B occurs if and only if the result of the experiment is 1, 3, or 5.

6 Intersection of Events, Union of Events, and Complement of an Event Let A and B be two events in the sample space S. The intersection of A and B, denoted by A B, is the set of all basic outcomes in S that belong to both A and B. The union of A and B, denoted by A B, is the set of all basic outcomes in S that belong to at least one of A and B. The complement of A, denoted by Ā or A, is the set of all basic outcomes in S that doesn t belong to A.

7 Mutually Exclusive Events Definition: If the events A and B have no common basic outcomes, they are called mutually exclusive and A B =. Note: For any event A, A and Ā are always mutually exclusive. Definition: Given the K events E 1, E 2,..., E K in the sample space S, if E 1 E 2 E K = S, these K events are said to be collectively exhaustive.

8 Example. A die is rolled. If events A, B, and C are defined as: A :"result is even", B :"result is at least 4", and C :"result is less than 6", describe the events a) A B b) A B C c) A d) A B e) A B C

9 Introduction Suppose that a random experiment is to be carried out and we want to determine the probability that a particular event will occur. Note: is measured over the range from 0 to 1. There are three definitions of probability: 1 Classical probability 2 Relative frequency probability 3 Subjective probability In this course, we will focus only on the first one.

10 Classical Definition: Classical probability is the proportion of times that an event will occur (assuming that all outcomes in a sample space are equally likely to occur). The probability of an event A is P (A) = n (A) n (S), where n (A) is the number of outcomes that satisfy the condition of event A and n (S) is the total number of outcomes in the sample space.

11 Example. If two dice are rolled, what is the probability that the sum of the upturned faces will equal 7? Solution. S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} n (S) = 36 A = sum of the upturned faces is 7 = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)} n (A) = 6 Therefore, P (A) = n(a) n(s) = 6 36 = 1 6 =

12 Counting, Permutations, and Combinations Generalized Basic Principle of Counting If r experiments that are to be performed are such that the first one may result in any of n 1 possible outcomes, and for each of these n 1 possible outcomes there are n 2 possible outcomes of the second experiment, and if for each of the possible outcomes of the first two experiments there are n 3 possible outcomes of the third experiment, and if..., then there is a total of n 1 n 2 n r possible outcomes of the r experiments.

13 Counting, Permutations, and Combinations Example. In a medical study patients are classified according to their blood types (A, B, AB, and 0) and according to their blood pressure levels (low, normal, and high). In how many different ways a patient can be classified?

14 Counting, Permutations, and Combinations Example. How many different outcomes are possible if a coin is tossed twice and a die is rolled?

15 Counting, Permutations, and Combinations Permutations and Combinations 1. Number of orderings We begin with the problem of ordering. Suppose that we have x objects that are to be placed in order in such a way that each object may be used only once. The total number of possible ways of arranging x objects in order is given by x (x 1) (x 2) (2) (1) = x!, where x! is read "x factorial".

16 Counting, Permutations, and Combinations Permutations and Combinations 2. Permutations Suppose that now we have n objects with which the x ordered boxes could be filled (n > x) in such a way that each object may be used only once. The total number of permutations of x object chosen from n, P n x, is the number of possible arrangements when x objects are to be selected from a total of n and arranged in order. This number is P n x = n (n 1) (n 2) (n x + 2) (n x + 1) = n! (n x)!.

17 Counting, Permutations, and Combinations Permutations and Combinations 3. Combinations Suppose that we are interested in the number of different ways that x can be selected from n (where no object may be chosen more than once) but the order is not important. The number of combinations of x object chosen from n, Cx n, is the number of possible selections that can be made. This number is (n x)! n! Cx n = Pn x x! = x! = n! (n x)!x! = ( ) n. x

18 Counting, Permutations, and Combinations Example. In how many different ways can a person invite three of her eight closest friends to a party?

19 Counting, Permutations, and Combinations Example. In how many ways can we choose three letters from A,B,C,D a) if the order is important? b) if the order is not important?

20 Counting, Permutations, and Combinations Example. In how many different ways can the letters in UNUSUALLY be arranged?

21 Counting, Permutations, and Combinations Example. In how many different ways can we arrange 10 books if 6 of them are on Mathematics, 3 are on Chemistry, 1 is on Physics?

22 Counting, Permutations, and Combinations Example. Consider a shuffled deck of 52 cards. a) In how many different ways can an Ace (A) be drawn? b) What is the probability of drawing an Ace (A)?

23 Counting, Permutations, and Combinations Example. There are five laptops of which three are brand A and two are brand B. Two of them will be chosen at random. a) Find the sample space. b) Define the event E by "One brand A and one brand B laptops will be chosen." and list the elements of event E. c) What is the probability that one brand A and one brand B laptops will be chosen? (Find P (E).)

24 Counting, Permutations, and Combinations Example. Suppose that there are ten brand A, five brand B, and four brand C laptops and three of them will be chosen at random. What is the probability that two of them will be brand A and one will be brand C?

25 Counting, Permutations, and Combinations Example. A manager is available a pool of 8 employees who could be assigned to a project-monitoring task. 4 of the employees are women and 4 are men. 2 of the men are brothers. The manager is to make the assignment at random so that each of the 8 employees is equally likely to be chosen. Let A be the event that "Chosen employee is a man." and B be the event that "Chosen employee is one of the brothers." a) Find P (A). b) Find P (B). c) Find P (A B).

26 Rules of probability are: 1 P (S) = 1 2 For any event A S, 0 P (A) 1 3 If A 1, A 2,... is a countable collection of mutually exclusive events, then P (A 1 A 2 ) = P (A 1 ) + P (A 2 ) + 4 For any event A S, P ( Ā ) = 1 P (A) 5 P ( ) = 0 6 For any two events A S and B S, P (A B) = P (A) + P (B) P (A B)

27 Example. The probability of A is 0.60, the probability of B is 0.45, and the probability of both is a) What is the probability of either A and B? b) What are P ( Ā ) and P ( B)?

28 Example. A pair of dice is rolled. If A is the event that a total of 7 is rolled and B is the event that at least one die shows up 4, find the probabilities for A, B, A B, and A B.

29 Example. A corporation has just received new machinery that must be installed and checked before it becomes operational. The accompanying table shows a manager s probability assessment for the number of days required before the machinery becomes operational. Number of days Let A be the event "It will be more than four days before the machinery becomes operational." and let B be the event "It will be less than six days before the machinery becomes available." a) P (A) =? b) P (B) =? c) P ( Ā ) =? d) P (A B) =? e) P (A B) =?

30 Conditional Suppose that two fair dice were rolled and we saw that one of them is 3. Under this condition, what is the probability of getting a total of 5? Since we know that one of the the dice is 3, we will be dealing with the restricted sample space S R = {(1, 3), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 3), (5, 3), (6, 3)} instead of the whole sample space S = {(1, 1), (1, 2),... (6, 6)} consisting of 36 basic outcomes. Therefore the desired event is {(2, 3), (3, 2)} and the desired probability is 2 11 =

31 Conditional Definition: Let A and B be two events. The conditional probability of event A, given that event B has occurred, is denoted by P (A B) and is defined as P (A B) = P (A B), provided that P (B) > 0. P (B) Similarly, P (B A) = P (A B), provided that P (A) > 0. P (A)

32 Example. An international cargo company knows that 75% of its customers prefer shipments with SMS support while 80% prefer shipments with internet support. It is also known that 65% of the customers prefer both. What are the probabilities that a) a customer who prefer SMS support will also prefer internet support? b) a customer who prefer internet support will also prefer SMS support?

33 Example. The probability that there will be a shortage of cement is 0.28 and the probability that there will not be a shortage of cement and a construction job will be finished on time is What is the probability that the construction job will be finished on time given that there will not be a shortage of cement?

34 Multiplication Rule Definition: Let A and B be two events. Using the definitions of conditional probabilities P (A B) and P (B A), we have P (A B) = P (A B) P (B) P (A B) = P (B A) P (A).

35 Example. What is the probability of getting 2 Aces (A) when two cards are drawn randomly from an ordinary deck of 52 cards?

36 Statistical Independence Definition: Let A and B be two events. A and B are said to be statistically independent if and only if P (A B) = P (A) P (B). More generally, the events E 1, E 2,..., E K are mutually statistically independent if and only if P (E 1 E 2 E K ) = P (E 1 ) P (E 2 ) P (E K ).

37 Statistical Independence If A and B are (statistically) independent, then and P (A B) = P (B A) = P (A B) P (B) P (A B) P (A) = = P (A) P (B) P (B) P (A) P (B) P (A) = P (A), provided that P (B) > 0 = P (B), provided that P (A) > 0.

38 Statistical Independence Note: Do not confuse independent events with mutually exclusive events: A and B are independent P (A B) = P (A) P (B) A and B are mutually exclusive A B = P (A B) = 0.

39 Example. Consider drawing a card from an ordinary deck of 52 cards. Let A :"Getting a queen (Q)" and B :"Getting a spade ( ). Are the events A and B independent?"

40 Example. Experience about a specific model of a mobile phone is that 80% of this model will operate for at least one year before repair is required. A director buys three of these phones. What is the probability that all three phones will work for at least one year without a problem?

41 Example. A quality-control manager found that 30% of worker-related problems occurred on Mondays and that 20% occurred in the last hour of a day s shift. It was also found that 4% of the worker-related problems occurred in the last hour of Monday s shift. a) What is the probability that a worker-related problem that occurs on a Monday does not occur in the last hour of the day s shift? b) Are the events "problem occurs on Monday" and "problem occurs in the last hour of a day s shift" statistically independent?

42 Consider two distinct sets of events A 1, A 2,..., A H and B 1, B 2,..., B K. The events A i, i = 1, 2,... H and B j, j = 1, 2,... K are mutually exclusive and collectively exhaustive within their sets but intersections A i B j can occur between all events from the two sets. The following table illustrates the outcomes of bivariate events: Note: If the probabilities of all intersections are known, then the whole probability structure of the random experiment is known and other probabilities of interest can be calculated.

43 Definition: The intersection probabilities P (A i B j ), i = 1, 2,... H and B j, j = 1, 2,... K, are called joint probabilities. The probabilities for individual events P (A i ), i = 1, 2,... H, and P (B j ), j = 1, 2,... K, are called marginal probabilities.

44 Example. A potential advertiser wants to know both income and other relevant characteristics of the audience for a particular television show. Families may be categorized, using A i, as to whether they regularly, occasionally, or never watch a particular series. In addition, they can be categorized, using B j, according to low-, middle-, and high-income subgroups. Find all marginal probabilities.

45 We can find any conditional probability P (A i B j ) and P (B j A i ) if we know all the joint probabilities. Some conditional probabilities for the previous example are: P (A 1 B 1 ) = P (A 1 B 1 ) = 0.04 P (B 1 ) 0.32 = P (A 2 B 1 ) = P (A 2 B 1 ) P (B 1 ) P (A 3 B 1 ) = P (A 3 B 1 ) P (B 1 ) = = = = P (B 2 A 1 ) = P (A 1 B 2 ) P (A 1 ) P (B 3 A 3 ) = P (A 3 B 3 ) P (A 3 ) = = = = 0.25

46 We can construct a tree diagram using marginal and conditional probabilities and then obtain the joint probabilities using multiplication rule. The tree diagram for the previous example is:

47 If E 1, E 2,..., E K are mutually exclusive and collectively exhaustive events, then for any event A with P (A) 0 and for any i = 1, 2,... K P (E i A) = P (A E i ) P (E i ) P (A E 1 ) P (E 1 ) + P (A E 2 ) P (E 2 ) + + P (A E K ) P (E K )

48 Example. A hotel rents cars for its guests from three rental agencies. It is known that 25% are from agency X, 25% are from agency Y, and 50% are from agency Z. If 8% of the cars from agency X, 6% from agency Y, and 15% from agency Z need tune-ups, what is the probability that a car needing a tune-up come from agency Y?

49 Example. A life insurance salesman finds that, of all the sales he makes, 70% are to people who already own policies. He also finds that, of all contacts for which no sale is made, 50% already own life insurance policies. Furthermore, 40% of all contacts result in sales. What is the probability that a sale will be made to a contact who already owns a policy?

50 Example. In a large city, 8% of the inhabitants have contracted a particular disease. A test for this disease is positive in 80% of people who have the disease and is negative in 80% of people who do not have the disease. What is the probability that a person for whom the test result is positive has the disease?

51 Example. A record-store owner assesses customers entering the store as high school age, college age, or older, and finds that of all customers 30%, 50%, and 20%, respectively, fall into these categories. The owner also found that purchases were made by 20% of high school age customers, by 60% of college age customers, and by 80% of older customers. a) What is the probability that a randomly chosen customer entering the store will make a purchase? b) If a randomly chosen customer makes a purchase, what is the probability that this customer is high school age?

52 Example. A restaurant manager classifies customers as regular, occasional, or new, and finds that of all customers 50%, 40%, and 10%, respectively, fall into these categories. The manager found that wine was ordered by 70% of the regular customers, by 50% of the occasional customers, and by 30% of the new customers. a) What is the probability that a randomly chosen customer orders wine? b) If wine is ordered, what is the probability that the person ordering is a regular customer? c) If wine is ordered, what is the probability that the person ordering is an occasional customer?

Chapter 1. Probability

Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

4.1 Sample Spaces and Events

4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an

Chapter 1. Probability

Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

Intermediate Math Circles November 1, 2017 Probability I

Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.

8.2 Union, Intersection, and Complement of Events; Odds

8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context

Contents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting - Permutation and Combination 39

CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting - Permutation and Combination 39 2.5

Section Introduction to Sets

Section 1.1 - Introduction to Sets Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase

7.1 Experiments, Sample Spaces, and Events

7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment

4.3 Rules of Probability

4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:

PROBABILITY. 1. Introduction. Candidates should able to:

PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation

Business Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal

Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter,

Math 227 Elementary Statistics. Bluman 5 th edition

Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical

CSC/MATA67 Tutorial, Week 12

CSC/MATA67 Tutorial, Week 12 November 23, 2017 1 More counting problems A class consists of 15 students of whom 5 are prefects. Q: How many committees of 8 can be formed if each consists of a) exactly

November 6, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern

STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes

STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle

Mathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015

1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:

Math 1313 Section 6.2 Definition of Probability

Math 1313 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

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability

Such a description is the basis for a probability model. Here is the basic vocabulary we use.

5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these

Applied Statistics I

Applied Statistics I Liang Zhang Department of Mathematics, University of Utah June 12, 2008 Liang Zhang (UofU) Applied Statistics I June 12, 2008 1 / 29 In Probability, our main focus is to determine

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even

, -the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.

4-1 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,

Probability. Ms. Weinstein Probability & Statistics

Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random

Probability and Counting Techniques

Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each

CHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events

CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes

Section : Combinations and Permutations

Section 11.1-11.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words

RANDOM EXPERIMENTS AND EVENTS

Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In day-to-day life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting

1. How to identify the sample space of a probability experiment and how to identify simple events

Statistics Chapter 3 Name: 3.1 Basic Concepts of Probability Learning objectives: 1. How to identify the sample space of a probability experiment and how to identify simple events 2. How to use the Fundamental

Probability (Devore Chapter Two)

Probability (Devore Chapter Two) 1016-351-01 Probability Winter 2011-2012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of Probability................................

Section 7.1 Experiments, Sample Spaces, and Events

Section 7.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with observable results. 1. Which of the follow are experiments? (a) Going into a room and turning on a light.

Lecture 6 Probability

Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times

3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,

Stat210 WorkSheet#2 Chapter#2

1. When rolling a die 5 times, the number of elements of the sample space equals.(ans.=7,776) 2. If an experiment consists of throwing a die and then drawing a letter at random from the English alphabet,

Probability. Engr. Jeffrey T. Dellosa.

Probability Engr. Jeffrey T. Dellosa Email: jtdellosa@gmail.com Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional

November 8, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol

Week 3 Classical Probability, Part I

Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability

Axiomatic Probability

Axiomatic Probability The objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance thtat A will occur.

Probability Rules. 2) The probability, P, of any event ranges from which of the following?

Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,

Probability Models. Section 6.2

Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example

TEST A CHAPTER 11, PROBABILITY

TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability

Probability. Dr. Zhang Fordham Univ.

Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!

Section 5.4 Permutations and Combinations

Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to

Probability Concepts and Counting Rules

Probability Concepts and Counting Rules Chapter 4 McGraw-Hill/Irwin Dr. Ateq Ahmed Al-Ghamedi Department of Statistics P O Box 80203 King Abdulaziz University Jeddah 21589, Saudi Arabia ateq@kau.edu.sa

Grade 6 Math Circles Fall Oct 14/15 Probability

1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.

Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Diana Pell Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 1. An experiment

I. 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

Section 5.4 Permutations and Combinations

Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! = n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to

CHAPTER 7 Probability

CHAPTER 7 Probability 7.1. Sets A set is a well-defined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can

Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)

12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the

Probability. Probabilty Impossibe Unlikely Equally Likely Likely Certain

PROBABILITY Probability The likelihood or chance of an event occurring If an event is IMPOSSIBLE its probability is ZERO If an event is CERTAIN its probability is ONE So all probabilities lie between 0

Probability and Randomness. Day 1

Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of

Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Before we can move from descriptive statistics to inferential statistics, we need to have some understanding of probability: Ch4: Probability and Counting Rules

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE

MATH 205 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM # - SPRING 2006 - DR. DAVID BRIDGE TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Tell whether the statement is

1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,

Textbook: pp Chapter 2: Probability Concepts and Applications

1 Textbook: pp. 39-80 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.

1. How many subsets are there for the set of cards in a standard playing card deck? How many subsets are there of size 8?

Math 1711-A Summer 2016 Final Review 1 August 2016 Time Limit: 170 Minutes Name: 1. How many subsets are there for the set of cards in a standard playing card deck? How many subsets are there of size 8?

Empirical (or statistical) probability) is based on. The empirical probability of an event E is the frequency of event E.

Probability and Statistics Chapter 3 Notes Section 3-1 I. Probability Experiments. A. When weather forecasters say There is a 90% chance of rain tomorrow, or a doctor says There is a 35% chance of a successful

ECON 214 Elements of Statistics for Economists

ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing

Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1

3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is

Introduction to Probability and Statistics I Lecture 7 and 8

Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes

Theory of Probability - Brett Bernstein

Theory of Probability - Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of

Simple Probability. Arthur White. 28th September 2016

Simple Probability Arthur White 28th September 2016 Probabilities are a mathematical way to describe an uncertain outcome. For eample, suppose a physicist disintegrates 10,000 atoms of an element A, and

MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG

MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices

Finite Mathematics MAT 141: Chapter 8 Notes

Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication

Statistics Intermediate Probability

Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting

Functional Skills Mathematics

Functional Skills Mathematics Level Learning Resource Probability D/L. Contents Independent Events D/L. Page - Combined Events D/L. Page - 9 West Nottinghamshire College D/L. Information Independent Events

Chapter 5 Probability

Chapter 5 Probability Math150 What s the likelihood of something occurring? Can we answer questions about probabilities using data or experiments? For instance: 1) If my parking meter expires, I will probably

12 Probability. Introduction Randomness

2 Probability Assessment statements 5.2 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. The probability of an event A as P(A) 5 n(a)/n(u ). The complementary events as

Chapter 5 - Elementary Probability Theory

Chapter 5 - Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling

4.4: The Counting Rules

4.4: The Counting Rules The counting rules can be used to discover the number of possible for a sequence of events. Fundamental Counting Rule In a sequence of n events in which the first one has k 1 possibilities

CHAPTERS 14 & 15 PROBABILITY STAT 203

CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical

Probability - Chapter 4

Probability - Chapter 4 In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. A cynical person

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

6. Practice Problems Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the probability. ) A bag contains red marbles, blue marbles, and 8

( ) Online MC Practice Quiz KEY Chapter 5: Probability: What Are The Chances?

Online MC Practice Quiz KEY Chapter 5: Probability: What Are The Chances? 1. Research on eating habits of families in a large city produced the following probabilities if a randomly selected household

Def: The intersection of A and B is the set of all elements common to both set A and set B

Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:

The next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:

CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such

Chapter 2. Permutations and Combinations

2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find

Chapter 1: Sets and Probability

Chapter 1: Sets and Probability Section 1.3-1.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping

Class XII Chapter 13 Probability Maths. Exercise 13.1

Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:

Beginnings of Probability I

Beginnings of Probability I Despite the fact that humans have played games of chance forever (so to speak), it is only in the 17 th century that two mathematicians, Pierre Fermat and Blaise Pascal, set

(a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?

Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent

4.1 What is Probability?

4.1 What is Probability? between 0 and 1 to indicate the likelihood of an event. We use event is to occur. 1 use three major methods: 1) Intuition 3) Equally Likely Outcomes Intuition - prediction based

CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information

November 11, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.

Elementary Statistics. Basic Probability & Odds

Basic Probability & Odds What is a Probability? Probability is a branch of mathematics that deals with calculating the likelihood of a given event to happen or not, which is expressed as a number between

Math 166: Topics in Contemporary Mathematics II

Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define

STAT 155 Introductory Statistics. Lecture 11: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 11: Randomness and Probability Model 10/5/06 Lecture 11 1 The Monty Hall Problem Let s Make A Deal: a game show

The topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:

CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Statistics Homework Ch 5 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability

Probability and Counting Rules. Chapter 3

Probability and Counting Rules Chapter 3 Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of

1) If P(E) is the probability that an event will occur, then which of the following is true? (1) 0 P(E) 1 (3) 0 P(E) 1 (2) 0 P(E) 1 (4) 0 P(E) 1

Algebra 2 Review for Unit 14 Test Name: 1) If P(E) is the probability that an event will occur, then which of the following is true? (1) 0 P(E) 1 (3) 0 P(E) 1 (2) 0 P(E) 1 (4) 0 P(E) 1 2) From a standard

Probability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College

Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical

Counting and Probability Math 2320

Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A

STOR 155 Introductory Statistics. Lecture 10: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10: Randomness and Probability Model 10/6/09 Lecture 10 1 The Monty Hall Problem Let s Make A Deal: a game show

Basic Probability Models. Ping-Shou Zhong

asic Probability Models Ping-Shou Zhong 1 Deterministic model n experiment that results in the same outcome for a given set of conditions Examples: law of gravity 2 Probabilistic model The outcome of the

Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules

+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that

Block 1 - Sets and Basic Combinatorics. Main Topics in Block 1:

Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.

Basic Probability Ideas. Experiment - a situation involving chance or probability that leads to results called outcomes.

Basic Probability Ideas Experiment - a situation involving chance or probability that leads to results called outcomes. Random Experiment the process of observing the outcome of a chance event Simulation