# Fundamentals of Probability

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 event would have a probability of 0; a certain event would have a probability of 1. This packet will focus on the foundation of probability and introduce different techniques for solving probability questions. When answering probability questions, use an exact fraction or round to 3 significant digits (leading zeros are not significant) unless told otherwise. Classical (Theoretical) Probability Formula For the Classical Probability Formula, the outcomes must be equally likely. If the outcomes are not equally likely, then the Empirical Probability Formula should be used.. Example: What is the probability of drawing a 7 from a standard deck of 52 cards? Solution: In a standard deck of cards, there are suits of cards each: Spades (black), Clubs (black), Hearts (red), and Diamonds (red). Each suit contains the following: the numbers through, Jack, Queen, King, and Ace. Therefore, a standard deck of cards contains four 7s. Empirical Probability Formula (Actual data) Note: The more times an experiment is repeated, the closer the Empirical probability will come to the theoretical probability. The question does not always provide the total number of outcomes. If the problem does not specify the total number of outcomes, there are different ways of finding this quantity depending on the situation. Finding Total Number of Outcomes When the number of choices is small, the creation of a table or other visual diagram can assist in determining the number of outcomes. Provided by Tutoring Services 1 Fundamentals of Probability Created December 2012

2 Example: What is the probability of a couple having exactly 3 girls in a family of 4? Assume the probability of having a girl is equally as likely as having a boy, and the gender of one child does not influence the gender of the others. Solution: The first step is to determine the total number of possible outcomes of children in a family of four. Since there are only two choices, boy or girl, we can find the number of total outcomes by evaluating, where the is the number of children in the family. Since the number is small, it is easy to make a chart of all the different combinations of four children. All the possible gender combinations a couple could have with four children are listed within the table to the right. There are 16 different gender combinations a couple can have with four children. Of those 16 ways, only 4 combinations result in exactly 3 girls. So the probability of a family having exactly three girls is: 1 st Child 2 nd Child 3 rd Child 4 th Child Boy Boy Boy Boy Boy Boy Boy Girl Boy Boy Girl Boy Boy Boy Girl Girl Boy Girl Boy Boy Boy Girl Boy Girl Boy Girl Girl Boy Boy Girl Girl Girl Girl Boy Boy Boy Girl Boy Boy Girl Girl Boy Girl Boy Girl Boy Girl Girl Girl Girl Boy Boy Girl Girl Boy Girl Girl Girl Girl Boy Girl Girl Girl Girl Sometimes it is not practical to use a table or other listing methods to find total number of outcomes. The following methods are often used to calculate the total number of outcomes. Counting Methods The diagram below can aid in determining which Counting Method is most appropriate for a question. Does the number of items equal the number of places avaiable? No Yes Can items be repeated? Use the Factorial Formula Yes No Use the Fundamental Counting Principal Is the order important? Yes No Use the Permutations Formua Use the Combinations Formula Provided by Tutoring Services 2 Fundamentals of Probability

3 Factorial Formula When arranging all of the items in the same number of places, use the Factorial Formula method. Example: Find the total number of ways 7 people can sit in 7 empty seats at a movie theater. Solution: To find the total number of ways 7 people can sit in 7 empty seats at a movie theater, consider filling one seat at a time. The first seat would be open to all 7 people. Once someone sits down, there are only 6 people left to sit in the next open seat. Then the next person sits down, and there are only 5 people left to sit in the next open seat. This process continues until there is only one person and one seat left. Mathematically, this example can be expressed as 7! (7 factorial) and is defined as: The total number of ways 7 people can sit in the 7 empty movie theater seats is:. Remember: You can use your TI 83/84 Plus to calculate the Factorial Formula: 1. First input value for total number of objects. 2. Press MATH 3. Use the Arrow Keys to highlight PRB 4. Use Arrow Keys to highlight 4:! 5. Press ENTER Fundamental Counting Principal (FCP) When repetitions are allowed and the number of ways to fill an open place is not affected by the way in which previous places are filled, the Fundamental Counting Principal method should be used. Multiply the number of ways each part of the task can be accomplished together. Example: A Virginia license plate consists of 3 letters followed by 4 numbers. If repeated letters and numbers are allowed, how many different license plates can be created? What is the probability of a license plate containing only one A? Solution: To find the total number of possible standard license plate options for the state of Virginia, first define the number of spaces on each plate. The standard plate consists of 3 letters followed by 4 numbers. Since there are 26 letters in the alphabet, there are 26 options for the first 3 spaces. The 4 number spots must be filled with a single digit 0-9, so there are 10 options for the next 4 spaces The total number of outcomes is: Provided by Tutoring Services 3 Fundamentals of Probability

4 Now, to find the number of license plates that contain only one A, look at each space again. If the A were to appear in the first letter space, there is only one letter choice for the first letter space. Now consider the next letter space. Since the problem specifies only one A on the license plate, the letter A is not repeated. Therefore, only 25 letters are available to choose from for the second and third letter spaces; the number spaces are unaffected The total number of license plates with only one A in the first space is: If you evaluate the one A located in the second or third space, you would get the same number of license plates as if the A were located in the first space. The total number of ways to have one A on a license place is: Therefore, the probability of having one A on a license plate is:. Permutations When replacements are not allowed, and the order in which the items rank is important, use the Permutations method. The formula for calculating permutations is: The n is the total number of items, and the r is the number of ways they are being selected. Example: The student body is electing the class President, Vice-President, Treasurer, and Secretary from a group of 14 students consisting of 5 seniors, 4 juniors, 3 sophomores, and 2 freshmen. What is the probability of electing all seniors? Solution: There are at least two ways to solve this problem. This solution will only cover the Permutations method. To begin we need to know how many different ways we can elect the class representatives. Since order matters to this problem, and the same person cannot be elected to two positions, use permutations. Provided by Tutoring Services 4 Fundamentals of Probability

5 Next we want to determine how many of the 24,024 arrangements consist of only seniors. Therefore, we want to determine how many ways we can arrange only the 5 seniors as the class representatives. The probability of electing all seniors as the class representatives is: In the following example, two different techniques will be needed to find the probability. Example: The student body is re-electing the class President, Vice-President, Treasurer, and Secretary from a group of 14 students consisting of 5 seniors, 4 juniors, 3 sophomores, and 2 freshmen. This time each class must be represented. What is the probability of electing one officer from each class? Solution: For this problem, analyze how many ways a member of each class can be elected and then how many ways we can arrange the four elected members in the positions similar to the Fundamental Counting Principal. How many ways can one member of each class be elected? The number ways a senior could be elected is: The number ways a junior could be elected is: The number ways a sophomore could be elected is: The number ways a freshman could be elected is: How many different ways can the four elected class members be arranged in the four offices? How many total ways can the class representatives be elected? Put all the information together to form the probability. The numerator will represent the number of ways one member of each class can be elected and arranged in the four offices. The denominator will represent the total number of ways the officers can be elected. 5 𝑃1 4 𝑃1 3 𝑃1 2 𝑃1 4 𝑃4 14 𝑃4 Use the TI 83/84 Plus to calculate Permutations: 1. Input value for n Provided by Tutoring Services 5 Fundamentals of Probability

6 2. Press MATH 3. Use the Arrow Keys to highlight PRB 4. Use the Arrow Keys to highlight 2: npr 5. Press ENTER 6. Input value for r 7. Press ENTER Combinations When replacements are not allowed and the order in which items are ranked does not matter, use the Combinations method. The formula for combinations is: Example: In order to win the Pick 5 lottery, a participant must select the correct 5 numbers in any order from between 1 and 52. What is the probability of winning the Pick 5 lottery? Solution: Since the order of the numbers does not matter, we can use combinations to determine the total possible number of combinations of 5 numbers from Since there is only one way to select the correct 5 numbers in any order, the probability of winning the Pick 5 lottery is: Use the TI 83/84 Plus to calculate Combinations: 1. Input value for n 2. Press MATH 3. Use the Arrow Keys to highlight PRB 4. Use the Arrow Keys to highlight 3: ncr 5. Press ENTER 6. Input value for r 7. Press ENTER Addition Rule A compound event combines two simple events. P (A or B) = P (event A occurs or event B occurs or they both occur). The formal addition rule is written as: Notice that the quantity is subtracted. This is because the events that fall under both event A and event B at the same time are counted twice: once when calculating, and then Provided by Tutoring Services 6 Fundamentals of Probability

7 again when calculating. If event A and event B are mutually exclusive, or cannot happen at the same time, the formula simplifies to: Example: What is the probability of selecting a test subject who received a negative test result or lied? Use the chart below. Positive Test Result (Test indicated subject lied) Negative Test Result (Test indicated subject did not lie) Subject did not lie Subject lied Solution: The total number of people tested is: To determine the number of successes, identify the two events in the probability. The two events are: subjects who lied and subjects who had negative test results. The number of subjects who lied is:. The number of subjects who received a negative test results is:.. Notice the is counted in both individual probabilities. Now apply the addition rule to get:. The Complement The complement of an event, A, consists of all outcomes where event A does not occur or. The sum of the probability of an event occurring and the probability of the event not occurring is equal to 1. This property is called the Rule of Complementary Events. The probability of event A not occurring is used in calculating the odds. Odds are always written in the most reduced terms. Provided by Tutoring Services 7 Fundamentals of Probability

8 Odds in favor: Odds against: favorable : unfavorable unfavorable : favorable Example: The Weather Channel predicted a 35% chance of rain today. What are the odds against rain? Solution: The probability of rain is 35%; the probability of no rain is 65%. The odds against rain are. Both 65 and 35 can be divided by 5. The final result is: The complement is also useful in solving several probability questions involving the phrase at least. The phrase at least one means one or more. Calculating the probability of an event occurring one time, two times, three times, four times, or n times can be long and involved. In these instances, calculating the complement can be faster. The complement of at least one is zero or none. Example: If a classroom contains 5 students, what is the probability that at least two of them have the same birthday? Solution: We start by determining the probability that none of the students have the same birthday. The probability of the first student having a birthday on any day is 365. The 365 probability of the next student not having the same birthday as the first student is 364. The 365 probability of the third student not having the same birthday as the first two students is 363, 365 and so on until we have all 5 students. The probability of at least two students having the same birthday is: Conditional Probability The conditional probability is the probability of event B occurring after it is assumed that event A has already occurred. The conditional probability is denoted:. Conditional probability is used with when the and indicates the probability of event A occurring followed by event B, not when and indicates the probability of event A and B happening at the same time. Conditional probability occurs without replacement. There are two types of conditional probabilities: dependent and independent. Events are considered dependent when the outcome of the first event affects the outcome of the second event. For example: drawing a heart from a standard card deck followed by a King. Provided by Tutoring Services 8 Fundamentals of Probability

9 Selecting a heart from the deck of cards and selecting a King from the deck of cards are dependent because if the King of Hearts is drawn first, there are now only three Kings left in the deck. For dependent events, the following conditional formula is used*: *If a sample is no more than % of the population, it should be treated as independent even though it is technically dependent. Events are considered independent when the outcome of the first event does not affect the outcome of the second event. For example: drawing a spade from a standard card deck followed by a diamond. When a Spade is removed from the deck, it does not affect the number of Diamonds remaining in the deck. In this case, the conditional formula above simplifies to: Example 1: What is the probability of drawing a Queen, a three, and then either a Queen or a three in that specific order? Assume there are no replacements. Solution 1: In a standard deck of cards, there are suits of cards each: Spades (black), Clubs (black), Hearts (red), and Diamonds (red). Each suit contains the following: the numbers through, Jack, Queen, King, and Ace. The probability of drawing a Queen for the first card is: Drawing a Queen and drawing a three are independent events since taking a Queen out of the deck does not affect the number of threes remaining in the deck. The denominator is because the first Queen drawn is not replaced. The probability of the second card being a three is: The probability of the third card being a Queen or a three is affected by the first two cards drawn and is therefore dependent. To calculate this probability, assume the first card drawn was a Queen, and the second card drawn was a three. How many Queens and threes are left in the deck? There are only Queens and threes. The probability of the third card being a Queen or a three is: Provided by Tutoring Services 9 Fundamentals of Probability

10 Finally, multiply the probabilities together to find the probability. Example 2: The local tire company produced a batch of 100 tires. Four of the tires are defective. What is the probability a customer randomly selects three good tires? Solution 2: First, determine the number of tires in the batch that are good. Since the number of tires to choose from changes every time a tire is selected, the events are dependent. Check the 5% guideline to see if the events can be treated as independent even though they are technically dependent. Since three tires are less than five tires, the boundary of the 5% guideline, the events can be treated as independent even though they are technically dependent. 3 ( ) This will be very close to the answer calculated by treating the events as dependent. Provided by Tutoring Services 10 Fundamentals of Probability

11 Practice Problems 1. A three digit number is created by selecting three random digits from the following set: Numbers beginning with 0 are not permitted. What is the probability that one randomly selected number generated from the above list is not a multiple of 5? 2. Three men and three women are waiting to interview for a job. If the candidates are called in for their interviews in a random order, what is the probability that all three women will be interviewed first? 3. A student does not want to be late for his final exam in Statistics, so he sets the alarm on three battery-operated alarm clocks. If each individual alarm clock has a 10% chance of failing, what is the probability of all three alarm clocks failing? 4. What is the probability of drawing three random cards from a standard 52-card deck in the following order: a Jack, a Spade, a red non-face card (2, 3, 4, 5, 6, 7, 8, 9, or 10)? 5. Five-card Draw is a game using a standard 52-card deck where the players each draw a hand of five cards. A royal flush, consisting of the Ace, King, Queen, Jack, and ten from the same suit, is the best hand a player can have. What is the probability that a player will randomly draw a royal flush? 6. If three people are selected at random from the blood type chart below, what is the probability that at least one of them will be type O? Blood Type Chart A B AB O Rh Rh What is the probability that a randomly selected person from the chart above will have blood type B given that his/her blood is Rh+? 8. Joanna, a Statistics student, sits down to take her final exam and realizes that she doesn t know the answers to any of the questions because she didn t study at all. If there are 10 multiple choice questions with the options of A through D for each, and on the grading scale a 70% is required for a passing C, what is the probability that Joanna will get a 70% on her test if she randomly bubbles in answers? Assume that all questions are of equal point value and there is an equal likelihood that any letter will be correct. Provided by Tutoring Services 11 Fundamentals of Probability

12 Solutions: Provided by Tutoring Services 12 Fundamentals of Probability

### Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical

### Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

### Combinations If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded

Combinations If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded If 5 sprinters compete in a race and the fastest 3 qualify for the relay

### Find the indicated probability. 1) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd.

Math 0 Practice Test 3 Fall 2009 Covers 7.5, 8.-8.3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. ) If a single

### 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

### 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

### AP Stats - Probability Review

AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

### Basic Probability Theory II

RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample

### EXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS

EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the

### 2.5 Conditional Probabilities and 2-Way Tables

2.5 Conditional Probabilities and 2-Way Tables Learning Objectives Understand how to calculate conditional probabilities Understand how to calculate probabilities using a contingency or 2-way table It

### Probabilities of Poker Hands with Variations

Probabilities of Poker Hands with Variations Jeff Duda Acknowledgements: Brian Alspach and Yiu Poon for providing a means to check my numbers Poker is one of the many games involving the use of a 52-card

### Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?

Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than

### Question of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay

QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know

### Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com

Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..

### 8.3 Probability Applications of Counting Principles

8. Probability Applications of Counting Principles In this section, we will see how we can apply the counting principles from the previous two sections in solving probability problems. Many of the probability

### Ready, Set, Go! Math Games for Serious Minds

Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -

### 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.

Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are

### A probability experiment is a chance process that leads to well-defined outcomes. 3) What is the difference between an outcome and an event?

Ch 4.2 pg.191~(1-10 all), 12 (a, c, e, g), 13, 14, (a, b, c, d, e, h, i, j), 17, 21, 25, 31, 32. 1) What is a probability experiment? A probability experiment is a chance process that leads to well-defined

### The study of probability has increased in popularity over the years because of its wide range of practical applications.

6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,

### Math 408, Actuarial Statistics I, Spring 2008. Solutions to combinatorial problems

, Spring 2008 Word counting problems 1. Find the number of possible character passwords under the following restrictions: Note there are 26 letters in the alphabet. a All characters must be lower case

### Section 6.2 Definition of Probability

Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will

### Statistics 100A Homework 2 Solutions

Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6

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

Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum

### Probability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space)

Probability Section 9 Probability Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) In this section we summarise the key issues in the basic probability

### Math 202-0 Quizzes Winter 2009

Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile

### (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

### 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

### Chapter 5 A Survey of Probability Concepts

Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible

### POKER LOTTO LOTTERY GAME CONDITIONS These Game Conditions apply, until amended or revised, to the POKER LOTTO lottery game.

POKER LOTTO LOTTERY GAME CONDITIONS These Game Conditions apply, until amended or revised, to the POKER LOTTO lottery game. 1.0 Rules 1.1 POKER LOTTO is governed by the Rules Respecting Lottery Games of

### Pure Math 30: Explained!

www.puremath30.com 323 Lesson 1, Part One: The Fundamental Counting Principle The Fundamental Counting Principle: This is an easy way to determine how many ways you can arrange items. The following examples

### Curriculum Design for Mathematic Lesson Probability

Curriculum Design for Mathematic Lesson Probability This curriculum design is for the 8th grade students who are going to learn Probability and trying to show the easiest way for them to go into this class.

### Math 728 Lesson Plan

Math 728 Lesson Plan Tatsiana Maskalevich January 27, 2011 Topic: Probability involving sampling without replacement and dependent trials. Grade Level: 8-12 Objective: Compute the probability of winning

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

Exam 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 that the result

### Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations)

Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations) Note: At my school, there is only room for one math main lesson block in ninth grade. Therefore,

### Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

### No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

### Determine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard.

Math 120 Practice Exam II Name You must show work for credit. 1) A pair of fair dice is rolled 50 times and the sum of the dots on the faces is noted. Outcome 2 4 5 6 7 8 9 10 11 12 Frequency 6 8 8 1 5

### Current California Math Standards Balanced Equations

Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2. (b) 1.5. (c) 0.5-2.

Stats: Test 1 Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the given frequency distribution to find the (a) class width. (b) class

### Worksheet A2 : Fundamental Counting Principle, Factorials, Permutations Intro

Worksheet A2 : Fundamental Counting Principle, Factorials, Permutations Intro 1. A restaurant offers four sizes of pizza, two types of crust, and eight toppings. How many possible combinations of pizza

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Exam. Name. How many distinguishable permutations of letters are possible in the word? 1) CRITICS

Exam Name How many distinguishable permutations of letters are possible in the word? 1) CRITICS 2) GIGGLE An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm,

### Chapter 3. Probability

Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.

### Chapter 4 - Practice Problems 1

Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula

### Homework 20: Compound Probability

Homework 20: Compound Probability Definition The probability of an event is defined to be the ratio of times that you expect the event to occur after many trials: number of equally likely outcomes resulting

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### Permutations & Combinations

Permutations & Combinations Extension 1 Mathematics HSC Revision Multiplication Rule If one event can occur in m ways, a second event in n ways and a third event in r, then the three events can occur in

### Section 6-5 Sample Spaces and Probability

492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

### PROBABILITY SECOND EDITION

PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All

### Hoover High School Math League. Counting and Probability

Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches

### Definition and Calculus of Probability

In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the

### Poker. 10,Jack,Queen,King,Ace. 10, Jack, Queen, King, Ace of the same suit Five consecutive ranks of the same suit that is not a 5,6,7,8,9

Poker Poker is an ideal setting to study probabilities. Computing the probabilities of different will require a variety of approaches. We will not concern ourselves with betting strategies, however. Our

### 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,

### 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

### Bayesian Tutorial (Sheet Updated 20 March)

Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that

### BEGINNER S BRIDGE NOTES. Leigh Harding

BEGINNER S BRIDGE NOTES Leigh Harding PLAYING THE CARDS IN TRUMP CONTRACTS Don t play a single card until you have planned how you will make your contract! The plan will influence decisions you will have

### Standard 12: The student will explain and evaluate the financial impact and consequences of gambling.

TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial

### Math 2001 Homework #10 Solutions

Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal

Table of Contents Introduction to Acing Math page 5 Card Sort (Grades K - 3) page 8 Greater or Less Than (Grades K - 3) page 9 Number Battle (Grades K - 3) page 10 Place Value Number Battle (Grades 1-6)

### Mathematics Higher Level

Mathematics Higher Level for the IB Diploma Exam Preparation Guide Paul Fannon, Vesna Kadelburg, Ben Woolley, Stephen Ward INTRODUCTION ABOUT THIS BOOK If you are using this book, you re probably getting

### Poker Probability from Wikipedia. Frequency of 5-card poker hands 36 0.00139% 0.00154% 72,192.33 : 1

Poker Probability from Wikipedia Frequency of 5-card poker hands The following enumerates the frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement.

### Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

### Lab 11. Simulations. The Concept

Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

### Homework 2 Solutions

CSE 21 - Winter 2012 Homework #2 Homework 2 Solutions 2.1 In this homework, we will consider ordinary decks of playing cards which have 52 cards, with 13 of each of the four suits (Hearts, Spades, Diamonds

### Methods Used for Counting

COUNTING METHODS From our preliminary work in probability, we often found ourselves wondering how many different scenarios there were in a given situation. In the beginning of that chapter, we merely tried

### 4. Binomial Expansions

4. Binomial Expansions 4.. Pascal's Triangle The expansion of (a + x) 2 is (a + x) 2 = a 2 + 2ax + x 2 Hence, (a + x) 3 = (a + x)(a + x) 2 = (a + x)(a 2 + 2ax + x 2 ) = a 3 + ( + 2)a 2 x + (2 + )ax 2 +

### MATH 105: Finite Mathematics 6-5: Combinations

MATH 105: Finite Mathematics 6-5: Combinations Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Developing Combinations 2 s of Combinations 3 Combinations vs. Permutations 4 Conclusion

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

### Normal Probability Distribution

Normal Probability Distribution The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use

### Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

### Statistics 100A Homework 1 Solutions

Chapter 1 tatistics 100A Homework 1 olutions Ryan Rosario 1. (a) How many different 7-place license plates are possible if the first 2 places are for letters and the other 5 for numbers? The first two

### PERMUTATIONS AND COMBINATIONS

PERMUTATIONS AND COMBINATIONS Mathematics for Elementary Teachers: A Conceptual Approach New Material for the Eighth Edition Albert B. Bennett, Jr., Laurie J. Burton and L. Ted Nelson Math 212 Extra Credit

### 7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School

7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School Page 1 of 20 Table of Contents Unit Objectives........ 3 NCTM Standards.... 3 NYS Standards....3 Resources

### Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

### 35 Permutations, Combinations and Probability

35 Permutations, Combinations and Probability Thus far we have been able to list the elements of a sample space by drawing a tree diagram. For large sample spaces tree diagrams become very complex to construct.

### MAT 155. Key Concept. February 03, 2011. 155S4.1 2_3 Review & Preview; Basic Concepts of Probability. Review. Chapter 4 Probability

MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 4 Probability 4 1 Review and Preview 4 2 Basic Concepts of Probability 4 3 Addition Rule 4 4 Multiplication Rule: Basics 4 7 Counting To find

### SECTION 10-5 Multiplication Principle, Permutations, and Combinations

10-5 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial

### UNDERGROUND TONK LEAGUE

UNDERGROUND TONK LEAGUE WWW.TONKOUT.COM RULES Players are dealt (5) five cards to start. Player to left of dealer has first play. Player must draw a card from the deck or Go For Low. If a player draws

### Using games to support. Win-Win Math Games. by Marilyn Burns

4 Win-Win Math Games by Marilyn Burns photos: bob adler Games can motivate students, capture their interest, and are a great way to get in that paperand-pencil practice. Using games to support students

### 2013 MBA Jump Start Program

2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

### The Procedures of Monte Carlo Simulation (and Resampling)

154 Resampling: The New Statistics CHAPTER 10 The Procedures of Monte Carlo Simulation (and Resampling) A Definition and General Procedure for Monte Carlo Simulation Summary Until now, the steps to follow

### Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin. You

### Standard 12: The student will explain and evaluate the financial impact and consequences of gambling.

STUDENT MODULE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Simone, Paula, and Randy meet in the library every

### LECTURE 3. Probability Computations

LECTURE 3 Probability Computations Pg. 67, #42 is one of the hardest problems in the course. The answer is a simple fraction there should be a simple way to do it. I don t know a simple way I used the

### 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

### 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

### In the situations that we will encounter, we may generally calculate the probability of an event

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

### 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

### Probability And Counting Problems with the TI-83

Probability And Counting Problems with the TI-83 By Bill Gallagher Grade 9/10 - Math A 5 day lesson plan TI-83 + Silver Graphing Calculator 1 Objectives for this unit - To create a better understanding

### Working with whole numbers

1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and

### Principles of Math 12 - Perms & Combs Practice Exam 1 www.math12.com

Principles of Math 1 - Perms & Combs Practice Exam 1 www.math1.com Permutations & Combinations Practice Exam Use this sheet to record your answers 1. NR 3. 17. 7.. 10. 18. 8. 3. NR 4. 19. 9. 4. 11. NR

### for the Bill Hanlon bill@hanlonmath.com

Strategies for Learning the Math Facts Bill Hanlon bill@hanlonmath.com The more sophisticated mental operations in mathematics of analysis, synthesis, and evaluation are impossible without rapid and accurate

### 7.S.8 Interpret data to provide the basis for predictions and to establish

7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an

### Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system

CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical

### 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

FUN + GAMES = MATHS Sue Fine Linn Maskell Teachers are often concerned that there isn t enough time to play games in maths classes. But actually there is time to play games and we need to make sure that