Lecture 6: Probability. If S is a sample space with all outcomes equally likely, define the probability of event E,

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lecture 6: Probability. If S is a sample space with all outcomes equally likely, define the probability of event E,"

Transcription

1 Lecture 6: Probability Example Sample Space set of all possible outcomes of a random process Flipping 2 coins Event a subset of the sample space Getting exactly 1 tail Enumerate Sets If S is a sample space with all outcomes equally likely, define the probability of event E, P(E) = the number of outcomes in E = n(e) the total number of outcomes in S n(s) ex. in flipping 2 coins, what s the probability of getting exactly 1 tail? ex. in flipping a coin what s the probability of getting a head? (What's E?) ex. in drawing a card, what s the probability of drawing a face card? Computing the Number of Possibilities When Order Counts ex. How many ways different 4-letter strings can be formed with the alphabet a,b,c? ex. How many 3-letter words can be formed? ex. How many ways can the Orioles and the Yankees win or lose a 3 game series? Lecture 13 p. 1

2 Multiplication Rule: If an operation consists of k steps step i k can be performed... in ways the entire operation can be performed in ways ex: How many user IDs are possible if each must consist of 5 characters and the first character must be an uppercase letter and the other 4 characters must be digits or uppercase letters? Some problems are not so simple: ex: How many results are there for the World Series? Game When Order Matters Using Trees to Determine the Number of Possible Outcomes (of a sequence of events) "Draw the tree and count the leaves" Example 1. How many sequences of 22 flips of a coin exist? Start Lecture 13 p. 2

3 Example 2: How many ways can you schedule your courses if CS can occur at 11 or 12, English at 10 or 2, and Physics at 10 or 1? Start [See the club officer example, pp ] Example 3: How many ways can a match in women s tennis proceed to its end? Lecture 13 p. 3

4 Permutations If all n distinct objects are rearranged in a sequence, this is called a permutation of the n objects. The number of permutations of n distinct elements is ex. How many ways can the 15 kindergarten children form a line? If only some, say r, of the n distinct objects are arranged in an ordered sequence, this is called an r-permutation. The number of r-permutations of a set of n distinct elements is ex. How many ways can 5 of the 15 kindergarten children be selected to form a line? Lecture 13 p. 4

5 Counting the Number of Elements in a Set (Remember, the elements in a set are NOT ordered.) ex. How many words are possible of length 4 or less? (assume only lowercase letters are used) Number of words of length 4 = Number of words of length 3 = ex. My grade report consists of 5 letter grades, one for each course, each an A, B, C, D, or F. How many possible grade reports will contain at least 1 A? Number of possible reports = Number of reports with at least 1 A= Suppose all the grade reports are equally likely. What s the probability that I ll get all A s? What is the probability of no A s? ex. I roll a die 3 times. How many possible sequences of rolls will contain at least 1 one? Number of possible sequences = Number of sequences with at least 1 one = What s the probability that I will roll at least 1 one? Lecture 13 p. 5

6 ex. In recent survey of 50 Loyola CS majors who can program in Java and/or C, 28 students said they can program only in Java and 12 said they can program only in C. How many can program in both? Inclusion/Exclusion Rule for Sets n(a B) = n(a) + n(b) n(a B) n(a B C ) = n(a) + n(b) + n(c) n(a B) n(a C) n(b C) + n (A B C) ex. In recent survey of 70 Loyola CS majors who can program in Java and/or C and/or Prolog, 18 students said they can program in Java and C and 10 said they can program in C and Prolog, 8 said they can program in Java and Prolog. 46 claimed to be able to program in Java, 38 in C, and 16 in Prolog. How many can program in all 3 languages? Lecture 13 p. 6

7 Combinations ex: Choose 2 representatives from our class for as CS council. How many different combinations are there? Does the order of the elements matter? In these examples, underline the "key" word that indicates that order does NOT matter. ex: How many pairs of socks are possible when selected from a drawer of 8 socks, each of a different color? ex: How many groups of 3 letters can be selected from the set { a, b, c, d }? ex: How many bridge foursomes are possible with members selected from this class? An r-combination of a set of n elements is a subset of r of the n elements. Denoted & n# $! % r " Lecture 13 p. 7

8 RELATIONSHIP BETWEEN PERMUTATIONS AND COMBINATIONS: Number of k-combinations = Number of k-permutations. The number of k-permutations containing the same set of elements & n # $! = % k " Read as " ex: How many groups of 6 students can I select from a group of 15 to ride in this car? ex: How many a groups of 4 students can be formed from a class of 3 females and 5 males in which exactly one of the students is female are possible? ex: Software Engineering teams of 4 members are being formed in a class of twenty students, of whom 6 are female and 14 are male. If all combinations are equally likely, what is the probability that a team will contain no females? Will contain exactly 1 female? Will contain at least 1 female? Will contain at most 1 female? Lecture 13 p. 8

Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.

Chapter 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 information

Coin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT.

Coin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? 2 How many ways can you get exactly 2 heads?

More information

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

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball. Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw

More information

Discrete Mathematics Lecture 5. Harper Langston New York University

Discrete Mathematics Lecture 5. Harper Langston New York University Discrete Mathematics Lecture 5 Harper Langston New York University Empty Set S = {x R, x 2 = -1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of

More information

Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850

Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850 Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do

More information

94 Counting Solutions for Chapter 3. Section 3.2

94 Counting Solutions for Chapter 3. Section 3.2 94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length-4 lists are there? Answer: 6 6 6 6 = 1296. (b

More information

Chapter 5 - Probability

Chapter 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 information

7 Probability. Copyright Cengage Learning. All rights reserved.

7 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 information

Jan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 50-54)

Jan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 50-54) Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0- Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample

More information

Chapter 6 Review 0 (0.083) (0.917) (0.083) (0.917)

Chapter 6 Review 0 (0.083) (0.917) (0.083) (0.917) Chapter 6 Review MULTIPLE CHOICE. 1. The following table gives the probabilities of various outcomes for a gambling game. Outcome Lose $1 Win $1 Win $2 Probability 0.6 0.25 0.15 What is the player s expected

More information

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

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 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 information

1 Combinations, Permutations, and Elementary Probability

1 Combinations, Permutations, and Elementary Probability 1 Combinations, Permutations, and Elementary Probability Roughly speaking, Permutations are ways of grouping things where the order is important. Combinations are ways of grouping things where the order

More information

Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.

Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52. Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive

More information

Probabilistic Strategies: Solutions

Probabilistic Strategies: Solutions Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

More information

33 Probability: Some Basic Terms

33 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 information

Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value. Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

More information

Section 2.1. Tree Diagrams

Section 2.1. Tree Diagrams Section 2.1 Tree Diagrams Example 2.1 Problem For the resistors of Example 1.16, we used A to denote the event that a randomly chosen resistor is within 50 Ω of the nominal value. This could mean acceptable.

More information

Expected Value 10/11/2005

Expected Value 10/11/2005 Expected Value 10/11/2005 Definition Let X be a numerically-valued discrete random variable with sample space Ω and distribution function m(x). The expected value E(X) is defined by E(X) = x Ω xm(x), provided

More information

Exam 1 Review Math 118 All Sections

Exam 1 Review Math 118 All Sections Exam Review Math 8 All Sections This exam will cover sections.-.6 and 2.-2.3 of the textbook. No books, notes, calculators or other aids are allowed on this exam. There is no time limit. It will consist

More information

1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9

1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9 400 lecture note #9 [Ch 9] Counting Methods 1 Introduction In many discrete problems, we are confronted with the problem of counting. Here we develop tools which help us counting. Examples: o [9.1.2 (p.

More information

MATH 3070 Introduction to Probability and Statistics Lecture notes Probability

MATH 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 information

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values. MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

More information

8.1: Sample Spaces, Events, and Probability

8.1: Sample Spaces, Events, and Probability 8.1: Sample Spaces, Events, and Probability 8.1.1 An experiment is an activity with observable results. An experiment that does not always give the same result, even under the same conditions, is called

More information

Chapter. Probability Pearson Education, Inc. All rights reserved. 1 of 20

Chapter. Probability Pearson Education, Inc. All rights reserved. 1 of 20 Chapter 3 Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 20 Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition

More information

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k. REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

More information

A1. Basic Reviews PERMUTATIONS and COMBINATIONS... or HOW TO COUNT

A1. Basic Reviews PERMUTATIONS and COMBINATIONS... or HOW TO COUNT A1. Basic Reviews Appendix / A1. Basic Reviews / Perms & Combos-1 PERMUTATIONS and COMBINATIONS... or HOW TO COUNT Question 1: Suppose we wish to arrange n 5 people {a, b, c, d, e}, standing side by side,

More information

A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.

A (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 information

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /

More information

P (A B) = P (AB)/P (B).

P (A B) = P (AB)/P (B). 1 Lecture 8 Conditional Probability Define the conditional probability of A given B by P (A B) = P (AB) P (B. If we roll two dice in a row the probability that the sum is 9 is 1/9 as there are four combinations

More information

Counting principle, permutations, combinations, probabilities

Counting 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 information

Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014 Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

More information

Sample Space, Events, and PROBABILITY

Sample 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 information

4 BASICS OF PROBABILITY. Experiment is a process of observation that leads to a single outcome that cannot be predicted with certainty.

4 BASICS OF PROBABILITY. Experiment is a process of observation that leads to a single outcome that cannot be predicted with certainty. 4 BASICS OF PROBABILITY Experiment is a process of observation that leads to a single outcome that cannot be predicted with certainty. Examples: 1. Pull a card from a deck 2. Toss a coin 3. Response time.

More information

In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events.

In 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. 4-2 Fundamentals Definitions:

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by

More information

MAT 1000. Mathematics in Today's World

MAT 1000. Mathematics in Today's World MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

More information

I. WHAT IS PROBABILITY?

I. 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 information

34 Probability and Counting Techniques

34 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 information

PROBABILITY 14.3. section. The Probability of an Event

PROBABILITY 14.3. section. The Probability of an Event 4.3 Probability (4-3) 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 information

Probability and Counting

Probability and Counting Probability and Counting Basic Counting Principles Permutations and Combinations Sample Spaces, Events, Probability Union, Intersection, Complements; Odds Conditional Probability, Independence Bayes Formula

More information

Math 30530: Introduction to Probability, Spring 2012

Math 30530: Introduction to Probability, Spring 2012 Name: Math 30530: Introduction to Probability, Spring 01 Midterm Exam I Monday, February 0, 01 This exam contains problems on 7 pages (including the front cover). Calculators may be used. Show all your

More information

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1

IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1 IEOR 4106: Introduction to Operations Research: Stochastic Models SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability Models, by Sheldon

More information

Math 166:505 Fall 2013 Exam 2 - Version A

Math 166:505 Fall 2013 Exam 2 - Version A Name Math 166:505 Fall 2013 Exam 2 - Version A On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work. Signature: Instructions: Part I and II are multiple choice

More information

For 2 coins, it is 2 possible outcomes for the first coin AND 2 possible outcomes for the second coin

For 2 coins, it is 2 possible outcomes for the first coin AND 2 possible outcomes for the second coin Problem Set 1. 1. If you have 10 coins, how many possible combinations of heads and tails are there for all 10 coins? Hint: how many combinations for one coin; two coins; three coins? Here there are 2

More information

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

More information

Discrete mathematics

Discrete mathematics Discrete mathematics Petr Kovář petr.kovar@vsb.cz VŠB Technical University of Ostrava DiM 470-2301/01, Winter term 2015/2016 About this file This file is meant to be a guideline for the lecturer. Many

More information

Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

More information

Statistics 100A Homework 3 Solutions

Statistics 100A Homework 3 Solutions Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win $ for each black ball selected and we

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.

**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 information

14.30 Introduction to Statistical Methods in Economics Spring 2009

14.30 Introduction to Statistical Methods in Economics Spring 2009 MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Lecture 11: Probability models

Lecture 11: Probability models Lecture 11: Probability models Probability is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model we need the following ingredients A sample

More information

AP Statistics 7!3! 6!

AP Statistics 7!3! 6! Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

More information

7.5: Conditional Probability

7.5: Conditional Probability 7.5: Conditional Probability Example 1: A survey is done of people making purchases at a gas station: buy drink (D) no drink (Dc) Total Buy drink(d) No drink(d c ) Total Buy Gas (G) 20 15 35 No Gas (G

More information

I. WHAT IS PROBABILITY?

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

More information

Name: Exam III. April 16, 2015

Name: Exam III. April 16, 2015 Department of Mathematics University of Notre Dame Math 10120 Finite Math Spring 2015 Name: Instructors: Garbett & Migliore Exam III April 16, 2015 This exam is in two parts on 10 pages and contains 15

More information

Toss a coin twice. Let Y denote the number of heads.

Toss a coin twice. Let Y denote the number of heads. ! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

More information

3. Conditional probability & independence

3. 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

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

More information

Math 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event

Math 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event Math 1320 Chapter Seven Pack Section 7.1 Sample Spaces and Events Experiments, Outcomes, and Sample Spaces An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment

More information

You flip a fair coin four times, what is the probability that you obtain three heads.

You flip a fair coin four times, what is the probability that you obtain three heads. Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

More information

Math 3C Homework 3 Solutions

Math 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 information

Most 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.

Most 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 information

If a tennis player was selected at random from the group, find the probability that the player is

If a tennis player was selected at random from the group, find the probability that the player is Basic Probability. The table below shows the number of left and right handed tennis players in a sample of 0 males and females. Left handed Right handed Total Male 3 29 32 Female 2 6 8 Total 4 0 If a tennis

More information

What is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts

What 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 information

Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes.

Odds: 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 information

7 Probability. Copyright Cengage Learning. All rights reserved.

7 Probability. Copyright Cengage Learning. All rights reserved. 7 Probability Copyright Cengage Learning. All rights reserved. 7.2 Relative Frequency Copyright Cengage Learning. All rights reserved. Suppose you have a coin that you think is not fair and you would like

More information

ECE-316 Tutorial for the week of June 1-5

ECE-316 Tutorial for the week of June 1-5 ECE-316 Tutorial for the week of June 1-5 Problem 35 Page 176: refer to lecture notes part 2, slides 8, 15 A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same

More information

Probabilities, Odds, and Expectations

Probabilities, Odds, and Expectations MATH 110 Week 8 Chapter 16 Worksheet NAME Probabilities, Odds, and Expectations By the time we are finished with this chapter we will be able to understand how risk, rewards and probabilities are combined

More information

Introductory Problems

Introductory Problems Introductory Problems Today we will solve problems that involve counting and probability. Below are problems which introduce some of the concepts we will discuss.. At one of George Washington s parties,

More information

Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

More information

Basic Probability Theory I

Basic 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 information

1. De Morgan s Law. Let U be a set and consider the following logical statement depending on an integer n. We will call this statement P (n):

1. De Morgan s Law. Let U be a set and consider the following logical statement depending on an integer n. We will call this statement P (n): Math 309 Fall 0 Homework 5 Drew Armstrong. De Morgan s Law. Let U be a set and consider the following logical statement depending on an integer n. We will call this statement P (n): For any n sets A, A,...,

More information

Probability OPRE 6301

Probability OPRE 6301 Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

More information

Probability. Vocabulary

Probability. 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 information

Probability Review. ICPSR Applied Bayesian Modeling

Probability Review. ICPSR Applied Bayesian Modeling Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How

More information

Module 6: Basic Counting

Module 6: Basic Counting Module 6: Basic Counting Theme 1: Basic Counting Principle We start with two basic counting principles, namely, the sum rule and the multiplication rule. The Sum Rule: If there are n 1 different objects

More information

Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability. 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 information

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,

More information

Math 421: Probability and Statistics I Note Set 2

Math 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 information

36 Odds, Expected Value, and Conditional Probability

36 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 information

7.5 Conditional Probability; Independent Events

7.5 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 information

NOTES: Justify your answers to all of the counting problems (give explanation or show work).

NOTES: Justify your answers to all of the counting problems (give explanation or show work). Name: Solutions Student Number: CISC 203 Discrete Mathematics for Computing Science Test 3, Fall 2010 Professor Mary McCollam This test is 50 minutes long and there are 40 marks. Please write in pen and

More information

Example. For example, if we roll a die

Example. 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 information

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

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sample Test 2 Math 1107 DeMaio Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Create a probability model for the random variable. 1) A carnival

More information

Minimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example

Minimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games

More information

Topic 1 Probability spaces

Topic 1 Probability spaces CSE 103: Probability and statistics Fall 2010 Topic 1 Probability spaces 1.1 Definition In order to properly understand a statement like the chance of getting a flush in five-card poker is about 0.2%,

More information

Deal or No Deal Lesson Plan

Deal or No Deal Lesson Plan Deal or No Deal Lesson Plan Grade Level: 7 (This lesson could be adapted for 6 th through 8 th grades) Materials: Deck of Playing Cards Fair Coin (coin with head and tail sides) for each pair of students

More information

Math 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. (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 information

Probability. 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? 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

Unit 18: Introduction to Probability

Unit 18: Introduction to Probability Unit 18: Introduction to Probability Summary of Video There are lots of times in everyday life when we want to predict something in the future. Rather than just guessing, probability is the mathematical

More information

A Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability - 1. Probability and Counting Rules

A 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 information

Events. Independence. Coin Tossing. Random Phenomena

Events. Independence. Coin Tossing. Random Phenomena Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,

More information

If 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

If 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 information

Alg2 Notes 7.4.notebook February 15, Two Way Tables

Alg2 Notes 7.4.notebook February 15, Two Way Tables 7 4 Two Way Tables Skills we've learned 1. Find the probability of rolling a number greater than 2 and then rolling a multiple of 3 when a number cube is rolled twice. 2. A drawer contains 8 blue socks,

More information

Math/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 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 information

7.1 Sample space, events, probability

7.1 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 information

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0. Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.

More information