# Honors Precalculus Chapter 9 Summary Basic Combinatorics

Size: px
Start display at page:

Download "Honors Precalculus Chapter 9 Summary Basic Combinatorics"

Transcription

1 Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each letter can be repeated? b) if 7 letters are used and no letter can be repeated? c) if 4 letters followed by 3 digits are used, no letter or digit can be repeated? d) if 7 letters or digits are used, each letter or digit can be repeated, and the first character must be a number? If a procedure P has a sequence of stages S 1, S 2,, S n and if S 1 can occur in r 1 ways S 2 can occur in r 2 ways S n can occur in r n ways Then the number of ways that the procedure P can occur is. C. Permutation: Definition: 1. How many ways can 4 people be lined up in a row for a photograph? 2. How many different words can be form using the word MATHEMATICS? The words don t have to make sense. Honors_Precalculus_Ch. 9_Summary pg 1 of 12

2 3. Sixteen people are trying out for roles as dwarfs in a production of Snow White and the Seven Dwarfs. In how many different ways can the director cast the seven roles? Permutation of an n-set: If all the elements of an n-set are distinguishable from one another, there are permutations. If an n-set contains n 1 objects of a first kind, n 2 object of a second kind, and so on with n 1 + n 2 + n k = n, then the number of distinguishable permutations of the n-set is If r objects are taken from a set of n (distinguishable) objects to be arranged in order (0 r n), the number of permutations is D. Combination: Definition: Example: 1. Five cards are selected from a deck of 52 cards to form a poker hand. How many different ways can this be done? 2. In the California Super Lotto Plus, five numbers are chosen from 1-47, no number is repeated and the order does not matter; also, one MEGA number is chosen from a) How many different tickets can be formed? b) You win a \$1 if your ticket matches three of the first five numbers. How many different tickets of this kind are possible? The number of combinations of n objects taken r at a time is Honors_Precalculus_Ch. 9_Summary pg 2 of 12

3 Binomial Theorem A. Pascal s Triangle Row 0: Row 1: Row 2: Row 3: Row 4: Describe the patterns in the Pascal s Triangle: Recursion Formula for Pascal s Triangle: " n% " \$ ' = % " \$ ' + % \$ ' or n C r = # r& # & # & B. Binomial Expansion: (a + b) 0 (a + b) 1 + (a + b) (a + b) (a + b) Describe the patterns in the Binomial Expansion, relate it to the Pascal s Triangle: Honors_Precalculus_Ch. 9_Summary pg 3 of 12

4 The Binomial Theorem: For any positive integer n (a + b) n = The rth term in the expansion of (a + b) n = Practice: 1. Expand the binomial: a) (x y) 7 b) (2x + y ) 5 c) Find the 5 th term in the expansion of (x 1) 6 Honors_Precalculus_Ch. 9_Summary pg 4 of 12

5 Probability A. Probability of an event: Define probability of an event in your own words. Give at least one example. What is the difference between empirical probability and theoretical probability: B. Visualizing Probability: Venn diagram & Tree diagram 1) In a high school, 54% of the students are girls and 62% of the students play sports. Half of the girls at the school play sports. Use a Venn diagram to answer the following questions: a) What percentage of the students who play sports are boys? b) If a student is chosen at random, what is the probability that it is a boy who does not play sports? 2. If it rains tomorrow, the probability is 60% that Mrs. Vu will stay home and watch a movie. If it does not rain tomorrow, there is a 10% chance that she will stay home and watch a movie. Suppose that the chance of rain tomorrow is 80%, what is the probability that Mrs. Vu will stay home and watch a movie? Use a tree diagram or a table. Honors_Precalculus_Ch. 9_Summary pg 5 of 12

6 C. Probability of independent events: 1) What is the probability that a family having four boys in a row? 2) Two dice are rolled. What is the probability of getting an even number on one die and a 1 on the other? Two events are independent if If A & B are two independent events then P(A and B) = p(a IB) = D. Probability of mutually exclusive events: 1) If a card is randomly selected from a standard deck of cards, a) what is the probability of getting a king or a queen? b) what is the probability of getting a king or a heart? 2) A group of people is comprised of 25 U.S. men, 20 U.S. women, 10 Canadian men, and 5 Canadian women. If a person is selected at random from the group, find the probability that the selected person is a man or a Canadian. Two events are mutually exclusive if If A & B are two events then P(A or B) = p(a U B) = If A & B are mutually exclusive then p(a IB) = so P(A or B) = p(a U B) = Honors_Precalculus_Ch. 9_Summary pg 6 of 12

7 E. Conditional Probability: 1) Two identical jars are on a table. Jar A contains 10 chocolate chip cookies and 5 peanut butter cookies. Jar B contains 5 chocolate chip cookies and 20 peanut butter cookies. What is the probability that a cookie selected at random is a chocolate chip cookie? 2) A particular drug test is 3% likely to produce a false positive result, that is, of 100 people that have not used drug were tested, the test reports would show 3 positive cases. The same drug test is 5% likely to produce a false negative result, that is, of 100 people that have used drug were tested, the test reports would show 5 negative cases. At a particular high school, the probability of a student having used drugs is 2%. What is the probability that a student tested positive actually used drugs? If event B is dependent on event A, then the probability of B occurring given that A occurred is p(b A) = Honors_Precalculus_Ch. 9_Summary pg 7 of 12

8 E. Binomial Distribution (Bernoulli principle) 1) A family has five children. a) What is the probability of having all boys? b) What is the probability of having exactly one boy? c) What is the probability of having exactly three boys? 2. Mrs. Vu gives her class a test of 15 multiple-choice questions. Each question has 4 answer choices, with only one correct choice. A student guesses randomly on all the questions. a) What is the probability of getting all 15 questions answered correctly? b) What is the probability of getting exactly 10 questions answered correctly? c) What is the probability of getting at least 10 questions answered correctly? Suppose an experiment consists of n independent repetitions of an experiment with two outcomes, called success and failure. Let p(success) = p and p(failure) = q. The probability of getting r successes out of n independent repetitions is Honors_Precalculus_Ch. 9_Summary pg 8 of 12

9 Sequences and Series A. Arithmetic Sequences: 1) a) Find the 100 th term of the arithmetic sequence: 1, 4, 7, 10, b) Define the value of the nth term of the above sequence explicitly. c) Define the value of the nth term of the above sequence recursively d) Find the sum of the first 100 terms. Arithmetic Sequence: A sequence {a n } is an arithmetic sequence if it can be written in the form Each term in an arithmetic sequence can be obtained by: Recursive formula: Explicit formula: Sum of a finite arithmetic sequence: Honors_Precalculus_Ch. 9_Summary pg 9 of 12

10 B. Geometric Sequences: 1) a) Find the 100 th term of the geometric sequence: 1, 2, 4, 8, 16, b) Define the value of the nth term of the above sequence explicitly. c) Define the value of the nth term of the above sequence recursively d) Find the series/sum of the first 100 terms. e) Find a formula for the sum of the first n terms of a geometric sequence. " e) Find # an for the sequence in part a. n=1 " f) Find # an for the sequence 1 n=1 2, 1 4, 1 8, Honors_Precalculus_Ch. 9_Summary pg 10 of 12

11 Geometric Sequence: A sequence {a n } is an arithmetic sequence if it can be written in the form Each term in an arithmetic sequence can be obtained by: Recursive formula: Explicit formula: Sum of a finite geometric series: Sum of an infinite geometric series for r < 1. C. Fibonacci sequence: Honors_Precalculus_Ch. 9_Summary pg 11 of 12

12 Mathematical Induction A. What is it for? B. Procedure for Mathematical Induction Proof: Let P n be a statement about the integer n. Step 1: Prove that P 1 is true. Step 2: Assume that P k is true, prove that P k+1 is true. 1. Prove that (2n 1) = n 2 is true for all positive integers n. Step 1: Show that the statement is true for n = 1. Step 2: Assume that the statement is true for n = k, show that the statement is true for n = k Prove that 3 is a factor of n 3 + 2n. Honors_Precalculus_Ch. 9_Summary pg 12 of 12

### Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY

Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY 1. Jack and Jill do not like washing dishes. They decide to use a random method to select whose turn it is. They put some red and blue

More information

### Unit Nine Precalculus Practice Test Probability & Statistics. Name: Period: Date: NON-CALCULATOR SECTION

Name: Period: Date: NON-CALCULATOR SECTION Vocabulary: Define each word and give an example. 1. discrete mathematics 2. dependent outcomes 3. series Short Answer: 4. Describe when to use a combination.

More information

### 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.

More information

### Review Questions on Ch4 and Ch5

Review Questions on Ch4 and Ch5 1. Find the mean of the distribution shown. x 1 2 P(x) 0.40 0.60 A) 1.60 B) 0.87 C) 1.33 D) 1.09 2. A married couple has three children, find the probability they are all

More information

### Elementary Combinatorics

184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are

More information

### MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology

MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally

More information

### 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.

More information

### The study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability

The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch

More information

### Name: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11

Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value

More information

### Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations

Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684)

More information

### Fundamental. If one event can occur m ways and another event can occur n ways, then the number of ways both events can occur is:.

12.1 The Fundamental Counting Principle and Permutations Objectives 1. Use the fundamental counting principle to count the number of ways an event can happen. 2. Use the permutations to count the number

More information

### 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

More information

### CSCI 2200 Foundations of Computer Science (FoCS) Solutions for Homework 7

CSCI 00 Foundations of Computer Science (FoCS) Solutions for Homework 7 Homework Problems. [0 POINTS] Problem.4(e)-(f) [or F7 Problem.7(e)-(f)]: In each case, count. (e) The number of orders in which a

More information

### 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

More information

### Week 1: Probability models and counting

Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model

More information

### Unit 14 Probability. Target 3 Calculate the probability of independent and dependent events (compound) AND/THEN statements

Target 1 Calculate the probability of an event Unit 14 Probability Target 2 Calculate a sample space 14.2a Tree Diagrams, Factorials, and Permutations 14.2b Combinations Target 3 Calculate the probability

More information

### a) 2, 4, 8, 14, 22, b) 1, 5, 6, 10, 11, c) 3, 9, 21, 39, 63, d) 3, 0, 6, 15, 27, e) 3, 8, 13, 18, 23,

Pre-alculus Midterm Exam Review Name:. Which of the following is an arithmetic sequence?,, 8,,, b),, 6, 0,, c), 9,, 9, 6, d), 0, 6,, 7, e), 8,, 8,,. What is a rule for the nth term of the arithmetic sequence

More information

### 9.3 Probability. ... and why Everyone should know how mathematical the laws of chance really are. EXAMPLE 1 Testing Your Intuition About Probability

658 CHAPTER 9 Discrete Mathematics 9.3 Probability What you ll learn about Sample Spaces and Probability Functions Determining Probabilities Venn Diagrams and Tree Diagrams Conditional Probability Binomial

More information

### Section 6.5 Conditional Probability

Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability

More information

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

More 9.-9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on

More information

### 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

More information

### DISCUSSION #8 FRIDAY MAY 25 TH Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics

DISCUSSION #8 FRIDAY MAY 25 TH 2007 Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics 2 Homework 8 Hints and Examples 3 Section 5.4 Binomial Coefficients Binomial Theorem 4 Example: j j n n

More information

### Section The Multiplication Principle and Permutations

Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different

More information

### Algebra II- Chapter 12- Test Review

Sections: Counting Principle Permutations Combinations Probability Name Choose the letter of the term that best matches each statement or phrase. 1. An illustration used to show the total number of A.

More information

### 9.1 Counting Principle and Permutations

9.1 Counting Principle and Permutations A sporting goods store offers 3 types of snowboards (all-mountain, freestyle, carving) and 2 types of boots (soft or hybrid). How many choices are there for snowboarding

More information

### Discrete Structures for Computer Science

Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #23: Discrete Probability Based on materials developed by Dr. Adam Lee The study of probability is

More information

### 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

More information

### Math 3201 Unit 3: Probability Name:

Multiple Choice Math 3201 Unit 3: Probability Name: 1. Given the following probabilities, which event is most likely to occur? A. P(A) = 0.2 B. P(B) = C. P(C) = 0.3 D. P(D) = 2. Three events, A, B, and

More information

### 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

More information

### Answer each of the following problems. Make sure to show your work.

Answer each of the following problems. Make sure to show your work. 1. A board game requires each player to roll a die. The player with the highest number wins. If a player wants to calculate his or her

More information

### Exam III Review Problems

c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous Week-in-Reviews

More information

### M146 - Chapter 5 Handouts. Chapter 5

Chapter 5 Objectives of chapter: Understand probability values. Know how to determine probability values. Use rules of counting. Section 5-1 Probability Rules What is probability? It s the of the occurrence

More information

### 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

More information

### Week in Review #5 ( , 3.1)

Math 166 Week-in-Review - S. Nite 10/6/2012 Page 1 of 5 Week in Review #5 (2.3-2.4, 3.1) n( E) In general, the probability of an event is P ( E) =. n( S) Distinguishable Permutations Given a set of n objects

More information

### LAMC Junior Circle February 3, Oleg Gleizer. Warm-up

LAMC Junior Circle February 3, 2013 Oleg Gleizer oleg1140@gmail.com Warm-up Problem 1 Compute the following. 2 3 ( 4) + 6 2 Problem 2 Can the value of a fraction increase, if we add one to the numerator

More information

### Math 1342 Exam 2 Review

Math 1342 Exam 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) If a sportscaster makes an educated guess as to how well a team will do this

More information

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

Ch. 3 Probability 3.1 Basic Concepts of Probability and Counting 1 Find Probabilities 1) A coin is tossed. Find the probability that the result is heads. A) 0. B) 0.1 C) 0.9 D) 1 2) A single six-sided

More 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.

More information

### Chapter 3: PROBABILITY

Chapter 3 Math 3201 1 3.1 Exploring Probability: P(event) = Chapter 3: PROBABILITY number of outcomes favourable to the event total number of outcomes in the sample space An event is any collection of

More information

### 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

More information

### Probability Theory. Mohamed I. Riffi. Islamic University of Gaza

Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 1 Probability Properties of probability Counting techniques 1 Chapter 1 Probability Probability Theorem P(φ)

More information

### Grade 7/8 Math Circles February 25/26, Probability

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely

More information

### 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

More information

### Math 1111 Math Exam Study Guide

Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the

More information

### 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

More information

### 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

More information

### 4. Are events C and D independent? Verify your answer with a calculation.

Honors Math 2 More Conditional Probability Name: Date: 1. A standard deck of cards has 52 cards: 26 Red cards, 26 black cards 4 suits: Hearts (red), Diamonds (red), Clubs (black), Spades (black); 13 of

More information

### 2. Combinatorics: the systematic study of counting. The Basic Principle of Counting (BPC)

2. Combinatorics: the systematic study of counting The Basic Principle of Counting (BPC) Suppose r experiments will be performed. The 1st has n 1 possible outcomes, for each of these outcomes there are

More information

### Fundamentals 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

### LC OL Probability. ARNMaths.weebly.com. As part of Leaving Certificate Ordinary Level Math you should be able to complete the following.

A Ryan LC OL Probability ARNMaths.weebly.com Learning Outcomes As part of Leaving Certificate Ordinary Level Math you should be able to complete the following. Counting List outcomes of an experiment Apply

More information

### Poker: Probabilities of the Various Hands

Poker: Probabilities of the Various Hands 22 February 2012 Poker II 22 February 2012 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13

More information

### 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

More information

### 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

More information

### PROBABILITY. Example 1 The probability of choosing a heart from a deck of cards is given by

Classical Definition of Probability PROBABILITY Probability is the measure of how likely an event is. An experiment is a situation involving chance or probability that leads to results called outcomes.

More information

### Counting Methods and Probability

CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You

More information

### 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

More information

### Permutations and Combinations Practice Test

Name: Class: Date: Permutations and Combinations Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Suppose that license plates in the fictional

More information

### Probability Review 41

Probability Review 41 For the following problems, give the probability to four decimals, or give a fraction, or if necessary, use scientific notation. Use P(A) = 1 - P(not A) 1) A coin is tossed 6 times.

More information

### 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

More information

### 10.1 Applying the Counting Principle and Permutations (helps you count up the number of possibilities!)

10.1 Applying the Counting Principle and Permutations (helps you count up the number of possibilities!) Example 1: Pizza You are buying a pizza. You have a choice of 3 crusts, 4 cheeses, 5 meat toppings,

More information

### Classical Definition of Probability Relative Frequency Definition of Probability Some properties of Probability

PROBABILITY Recall that in a random experiment, the occurrence of an outcome has a chance factor and cannot be predicted with certainty. Since an event is a collection of outcomes, its occurrence cannot

More information

### SALES AND MARKETING Department MATHEMATICS. Combinatorics and probabilities. Tutorials and exercises

SALES AND MARKETING Department MATHEMATICS 2 nd Semester Combinatorics and probabilities Tutorials and exercises Online document : http://jff-dut-tc.weebly.com section DUT Maths S2 IUT de Saint-Etienne

More information

### POKER (AN INTRODUCTION TO COUNTING)

POKER (AN INTRODUCTION TO COUNTING) LAMC INTERMEDIATE GROUP - 10/27/13 If you want to be a succesful poker player the first thing you need to do is learn combinatorics! Today we are going to count poker

More information

### 2. The figure shows the face of a spinner. The numbers are all equally likely to occur.

MYP IB Review 9 Probability Name: Date: 1. For a carnival game, a jar contains 20 blue marbles and 80 red marbles. 1. Children take turns randomly selecting a marble from the jar. If a blue marble is chosen,

More information

### North Seattle Community College Winter ELEMENTARY STATISTICS 2617 MATH Section 05, Practice Questions for Test 2 Chapter 3 and 4

North Seattle Community College Winter 2012 ELEMENTARY STATISTICS 2617 MATH 109 - Section 05, Practice Questions for Test 2 Chapter 3 and 4 1. Classify each statement as an example of empirical probability,

More information

### 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

More information

### 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

More information

### 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

More information

### Math 1111 Math Exam Study Guide

Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the

More information

### Poker: Probabilities of the Various Hands

Poker: Probabilities of the Various Hands 19 February 2014 Poker II 19 February 2014 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13

More information

### 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

More information

### Combinatorial Proofs

Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A

More information

### 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:

More information

### Discrete probability and the laws of chance

Chapter 8 Discrete probability and the laws of chance 8.1 Multiple Events and Combined Probabilities 1 Determine the probability of each of the following events assuming that the die has equal probability

More information

### 11.3B Warmup. 1. Expand: 2x. 2. Express the expansion of 2x. using combinations. 3. Simplify: a 2b a 2b

11.3 Warmup 1. Expand: 2x y 4 2. Express the expansion of 2x y 4 using combinations. 3 3 3. Simplify: a 2b a 2b 4. How many terms are there in the expansion of 2x y 15? 5. What would the 10 th term in

More information

### 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

More information

### 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

More information

### Finite Math B, Chapter 8 Test Review Name

Finite Math B, Chapter 8 Test Review Name Evaluate the factorial. 1) 6! A) 720 B) 120 C) 360 D) 1440 Evaluate the permutation. 2) P( 10, 5) A) 10 B) 30,240 C) 1 D) 720 3) P( 12, 8) A) 19,958,400 B) C)

More information

### STAT 311 (Spring 2016) Worksheet: W3W: Independence due: Mon. 2/1

Name: Group 1. For all groups. It is important that you understand the difference between independence and disjoint events. For each of the following situations, provide and example that is not in the

More information

### Section Summary. Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning

Section 7.1 Section Summary Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning Probability of an Event Pierre-Simon Laplace (1749-1827) We first study Pierre-Simon

More information

### CS 237: Probability in Computing

CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 5: o Independence reviewed; Bayes' Rule o Counting principles and combinatorics; o Counting considered

More information

### 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,

More information

### 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

More information

### 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

### Mutually Exclusive Events

Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number and you roll a 2? Can these both occur at the same time? Why or why not? Mutually

More information

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Chapter 3: Practice SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) A study of 000 randomly selected flights of a major

More information

### 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

More information

### 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

More information

### Counting in Algorithms

Counting Counting in Algorithms How many comparisons are needed to sort n numbers? How many steps to compute the GCD of two numbers? How many steps to factor an integer? Counting in Games How many different

More information

1 of 5 7/16/2009 6:57 AM Virtual Laboratories > 13. Games of Chance > 1 2 3 4 5 6 7 8 9 10 11 3. Simple Dice Games In this section, we will analyze several simple games played with dice--poker dice, chuck-a-luck,

More information

### 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

More information

### Section 8.1. Sequences and Series

Section 8.1 Sequences and Series Sequences Definition A sequence is a list of numbers. Definition A sequence is a list of numbers. A sequence could be finite, such as: 1, 2, 3, 4 Definition A sequence

More information

### Probability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability

Most people think they understand odds and probability. Do you? Decision 1: Pick a card Decision 2: Switch or don't Outcomes: Make a tree diagram Do you think you understand probability? Probability Write

More information

### Strings. A string is a list of symbols in a particular order.

Ihor Stasyuk Strings A string is a list of symbols in a particular order. Strings A string is a list of symbols in a particular order. Examples: 1 3 0 4 1-12 is a string of integers. X Q R A X P T is a

More information

### The probability set-up

CHAPTER 2 The probability set-up 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample

More information

### 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

More information

### 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

More information

### CS1802 Week 3: Counting Next Week : QUIZ 1 (30 min)

CS1802 Discrete Structures Recitation Fall 2018 September 25-26, 2018 CS1802 Week 3: Counting Next Week : QUIZ 1 (30 min) Permutations and Combinations i. Evaluate the following expressions. 1. P(10, 4)

More information

### Unit 9: Probability Assignments

Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose

More information

### Bell Work. Warm-Up Exercises. Two six-sided dice are rolled. Find the probability of each sum or 7

Warm-Up Exercises Two six-sided dice are rolled. Find the probability of each sum. 1. 7 Bell Work 2. 5 or 7 3. You toss a coin 3 times. What is the probability of getting 3 heads? Warm-Up Notes Exercises

More information