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

Size: px
Start display at page:

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

Transcription

1 Probability and Statistics Chapter 3 Notes Section 3-1 I. Probability Experiments. A. When weather forecasters say There is a 90% chance of rain tomorrow, or a doctor says There is a 35% chance of a successful surgery, they are stating the likelihood, or, that a specific event will occur. 1. Decisions such as should you plan a picnic for tomorrow, or should you proceed with surgery are almost always based on these probabilities. B. A probability experiment is an, or, through which specific results (, or ) are obtained. 1. The result of a single trial in a probability experiment is an. 2. The set of all possible outcomes of a probability experiment is the. 3. An is a of the sample space. a) An event may consist of one or more. C. Simple Example of the use of the terms probability experiment, sample space, event, and outcome. 1. Probability experiment:. 2. Sample Space: 3. Event: 4. Outcome: Section 3-1 Example 1: A probability example consists of tossing a coin and then rolling a six-sided die. Determine the number of outcomes and identify the sample space. There are possible outcomes when tossing a coin; a, or a. For each of these, there are possible outcomes when rolling a die:. One way to list all the possible outcomes (the sample space) for actions occurring in a sequence is to use a. D. Events are often represented by letters, such as and. 1. An event that consists of a single outcome is called a simple event. a) In Example 1, the event tossing heads and rolling a 3 is a simple event and can be represented as A = b) In contrast, the event tossing heads and rolling an even number is NOT simple because it consists of possible outcomes: B = Section 3-1 Example 2: Determine the number of outcomes in each event. Then decide whether each event is simple or not. 1) For quality control, you randomly select a machine part from a batch that has been manufactured that day. Event A is selecting a specific defective machine part. 2) You roll a six-sided die. Event B is rolling at least a 4. Event A event; you either pick that specific defective part or you don t Event B event; it has possible outcomes. You ask for a student s age at his or her last birthday. Decide whether each event is simple or not. 1) Event C: The student s age is between 15 and 18, inclusive. 2) Event D: The student s age is 17. Event C is ; the ages of 15, 16, 17 and 18 are all possible outcomes. Event D is : the only possible successful outcome is that the student is 17 years old.

2 II. The Fundamental Counting Principle A. If one event can occur in m ways and a second event can occur in n ways, the number of ways that the two events can occur in sequence is. 1) This rule can be extended for of events occurring in. B. In plain English, the number of ways that events can occur in sequence is found by the number of ways can occur by the number of ways the other event(s) can occur. Section 3-1 Example 3: You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed. Manufacturer: Ford, GM, Honda Car size: Compact, midsize Color: White (W), red (R), black (B), and green (G) How many different ways can you select one manufacturer, one car size, and one color? Use a tree diagram to check your answer. There are 3 manufacturers, 2 sizes, and 4 colors, so the number of different ways is equal to * *, or. Section 3-1 Example 4: The access code for a car s security system consists of 4 digits. Each digit can be 0 through 9. 1) How many access codes are possible if each digit can be used only once, and not repeated? 2) How many access codes are possible if each digit can be repeated? 3) How many access codes are possible if each digit can be repeated, but the first digit cannot be 0 or 1? SOLUTIONS: 1) There are possible choices for the first digit ( ) There are possible choices for the second digit. There are possible choices for the third digit. There are possible choices for the fourth digit. * * * = possible codes. 2) There are possible choices for the first digit ( ) There are possible choices for the second digit. There are possible choices for the third digit. There are possible choices for the fourth digit. * * * = possible codes. 3) There are possible choices for the first digit ( ) There are possible choices for the second digit. There are possible choices for the third digit. There are possible choices for the fourth digit. * * * = possible codes. III. Types of Probability A. Classical (or probability) is used when each outcome in a sample space is likely to occur. The classical probability for an event E is given by P(E) = Section 3-1 Example 5: You roll a six-sided die. Find the probability of each event. 1) Event A: rolling a 3 2) Event B: rolling a 7 3) Event C: rolling a number less than 5 number of outcomes in event E Total number of outcomes in sample space.

3 SOLUTIONS: 1) The sample space consists of outcomes: There is outcome in Event A: A = So, P (rolling a 3) = 2) The sample space consists of 6 outcomes: {1, 2, 3, 4, 5, 6} There are outcomes in Event B: So, P (rolling a 7) = 2) The sample space consists of 6 outcomes: {1, 2, 3, 4, 5, 6} There are outcomes in Event B: So, P (rolling < 5) = B. Empirical (or statistical) probability is used when each outcome of an event is likely to occur. When an experiment is repeated times, regular patterns are formed. These patterns make it possible to find probability. Empirical (or statistical) probability) is based on obtained from. The empirical probability of an event E is the frequency of event E. Frequency of event E P(E) = Total frequency n Section 3-1 Example 6: A company is conducting an online survey of randomly selected individuals to determine if traffic congestion is a problem in their community. So far, 320 people have responded to the survey. The frequency distribution shows the results. What is the probability that the next person that responds to the survey says that traffic congestion is a serious problem in their community? The event is a response of It is a serious problem. The frequency of this event is. Because the total of the frequencies is, the empirical probability of the next person saying that traffic congestion is a serious problem in their community is: P(Serious problem) = Frequency of event "It is a serious problem" Total frequency = C. probability: these result from intuition, educated guesses and estimates. They are not as trusted in statistical studies as theoretical or empirical probabilities are. D. Law of Large Numbers A. As an experiment is repeated over and over, the empirical probability of an event the ( ) probability of an event. If you toss a fair coin times, you may only get heads. If you toss a fair coin times, you will get very close to heads. Section 3-1 Example 7: You survey a sample of 1000 employees at a large company and record the age of each. The results are shown in the frequency distribution. If you randomly select another employee, what is the probability that the employee will be between 25 and 34 years old? Ages 15 to to to to to and over Total Frequencies

4 The event is selecting an employee who is between 25 and 34 years old. In your survey, the frequency of this event is 366. Because the total of the frequencies is 1000, the probability of selecting an employee between the ages of 25 and 34 years old is: P(25 to 34 years old) = Frequency of event "25 to 34 years old" Total frequency = Section 3-1 Example 8: Classify each statement as an example of classical probability, empirical probability, or subjective probability 1. The probability that you will be married by age 30 is The probability that a voter chosen at random will vote Republican is The probability of winning a 1000-ticket raffle with one ticket is SOLUTIONS 1. This is a probability, based on a or a. 2. This is an example of an probability, since it most likely resulted from a or and the outcomes are likely. 3. This is an example of ( ) probability. You know the number of outcomes, and each one is likely to occur. E. Range of Probabilities Rule 1. Very important to remember this AT ALL TIMES!! ALL probabilities have a value between and. An impossible event has a probability of. An event that is guaranteed to occur has a probability of. An event that has an equal chance of occurring or not occurring has a probability of. Any event with a probability of less than or greater than is considered to be unusual when it occurs. F. Complementary Events 1. Complementary events have probabilities that add up to. If there is a 76% chance of rain, and a 24% chance that it doesn t rain, raining or not raining are events. 2. Definition: The complement of event E is the set of all in a sample space that are not in event E. The complement of event E is denoted by E and is read as E prime. Section 3-1 Example 9: Use the frequency distribution in Example 7 to find the probability of randomly choosing an employee who is not between 25 and 34 years old. We already know that the probability of an employee being between 25 and 34 years old is. We also know that the event not between 25 and 34 is the of between 25 and 34. So, we can subtract from to get the probability that an employee is not between 25 and 34 years old. =. Section 3-2 I. Conditional Probability A conditional probability is the probability of an event occurring, given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is denoted P(B A) and is read as probability of B, given A. II. Independent and Dependent Events Two events are if the occurrence of one of the events affect the probability of the occurrence of the other event. Two events A and B are if P(B A) = P(B), or if P(A B) = P(A)

5 In other words, event B is likely to occur whether event A has or not. Events that are not independent are. To determine if A and B are independent, first calculate, the probability of event. Then calculate, the probability of, given. If the two probabilities are, the events are. If the two probabilities are not, the events are. III. The Multiplication Rule P(A and B) = P(A) P(B A) If Events A and B are dependent; 1) Find the probability the first event occurs 2) Find the probability the second event occurs given the first event has occurred. 3) Multiply these two probabilities to find the probability that both events will occur in sequence. If Events A and B are independent, P(A and B) = P(A) P(B). This simplified rule can be extended for any number of independent events, just like the Fundamental Counting Principle could be extended. Example 1 (Page 149) 1) Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. Assume that the king is not replaced (it s not put back into the deck before the queen is drawn. 2) The table shows the results of a study in which researchers examined a child s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ given that the child has the gene. Gene Present Gene Not Present Total High IQ Normal IQ Total SOLUTIONS: 1) The probability of drawing a queen after a king (or any other card) has been taken out of the deck is, or about. 2) There are children who have the gene. They are our. Of these, have a high IQ. So, P(B A) =. Example 2 (Page 150) Decide whether the events are independent or dependent. 1) Selecting king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B). 2) Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B). SOLUTIONS: 1) The probability of pulling a queen out of the deck is. The probability of pulling a queen out of the deck after a king has already been removed is. Since these probabilities are, the events are. 2) The probability of obtaining a 6 is. The probability of obtaining a 6 given that the coin came up heads is. Since these probabilities are, the events are. Example 3 (Page 151) 1) Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen. 2) A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6. SOLUTIONS: 1) Because the first card is not replaced, the events are. P(K and Q) = * * = 2) The two events are. P(H and 6) = P(H) P(6) * = Example 4 (Page 152) 1) A coin is tossed and a die is rolled. Find the probability of getting a tail and then rolling a 2.

6 2) The probability that a particular knee surgery is successful is Find the probability that three surgeries in a row are all successful. 3) Find the probability that none of the three knee surgeries is successful. 4) Find the probability that at least one of the three knee surgeries is successful. SOLUTIONS 1) The probability of getting a tail and then rolling a 2 is P(T and 2) = P(T) P(2) * = 2) The probability that each knee surgery is successful is Since these events are, the probability that all three are successful is found by their probabilities together. The probability that all three surgeries are successful is about. 3) Because the probability of success for one surgery is.85, the probability of failure for one surgery is. This is true because failure is the of success. The probability that each knee surgery is not successful is. Since these events are, the probability that all three are not successful is found by their probabilities together. The probability that none of the surgeries are successful is about. 4) The phrase at least one means. The complement to the event at least one is successful is the event. Using the complement rule, we can simply the probability that none were successful from to find the probability that at least one was successful. =. The probability that at least one of the surgeries is successful is about. Example 5 (Page 153) More than 15,000 medical school seniors applied to residency programs in Of those, 93% were matched to a residency position. 74% percent of the seniors matched to a residency position were matched to one of their top two choices. 1) Find the probability that a randomly selected senior was matched to a residency program and it was one of the senior s top two choices. 2) Find the probability that a randomly selected senior that was matched to a residency program did not get matched with one of the senior s top two choices. 3) Would it be unusual for a randomly selected senior to result in a senior that was matched to a residency position and it was one of the senior s top two choices? 1) These two events are : P(A and B) = P(A) P(B A) = *. 2) To find this probability, use the complement: P(B A) = 1 P(B A) = 1 - =. 3) : This event occurs around % of the time. Section 3-3 Mutually exclusive events are events that both happen at the time. The Addition Rule (For OR probabilities) Or can mean one of three things: A occurs and B does not occur. B occurs and A does not occur. Both A and B occur. If A and B are mutually exclusive, simply their probabilities of occurring together. If A and B are NOT mutually exclusive, add their probabilities of occurring together and then subtract the probability that.

7 Type of Probability & Rules Classical Probability Empirical Probability Range of Probabilities Rule Complementary Events Multiplication Rule Addition Rule In Words The number of outcomes in the sample space is known and each outcome is equally likely to occur The frequency of outcomes in the sample space is estimated from experimentation (you have data) ALL probabilities are between zero and 1, inclusive The complement of event E is the set of all outcomes in a sample space that are NOT included in E, denoted E The Multiplication Rule is used to find the probability of two events occurring in a sequence (AND) The Addition Rule is used to find the probability of at least one of two events In Symbols Number of outcomes in E P(E) = Number of outcomes in sample space Frequency of E Total Frequency = Frequency of E Total Frequency = f n 0 P(E) 1 P(E ) = 1 P(E); P(E) = 1 P(E ) P(A and B) = P(A) P(B A) (dependent) P(A and B) = P(A) P(B) (independent) P(A or B) = P(A) + P(B) P(A and B) P(A or B) = P(A) + P(B) (mutually exclusive) occurring (OR) Example 1: Determine whether these two events are mutually exclusive or not. 1) Roll 3 on a die AND roll 4 on a die. 2) Randomly select a male student AND randomly select a nursing major. 3) Randomly select a blood donor with type O blood AND randomly select a female blood donor. 4) Randomly select a jack from a deck of cards AND randomly select a face card from a deck of cards. 5) Randomly select a 20-year-old student AND randomly select a student with blue eyes. 6) Randomly select a vehicle that is a Ford AND randomly select a vehicle that is a Toyota. SOLUTIONS 1) You roll and 3 AND a 4 on the same roll. These are. 2) You be both a male and a nursing major. These are. 3) You be both a female and have type 0 blood. These are. 4) Jacks are face cards, so any jack is also a face card. These are. 5) You be a 20-year-old student and have blue eyes. These are. 6) A vehicle be a Ford and a Toyota at the same time. These are. Example 2: 1) You select a card from a standard deck. Find the probability that the card is a 4 or an ace. 2) You select a card from a standard deck. Find the probability that the card is a queen or a red card. SOLUTIONS 1) Ace and 4 are. the probability of getting an ace to the probability of getting a 4. The probability of getting an ace is ; the probability of getting a four is also. The probability of getting an ace or a four is = 8 52 = ) The events are (there are red queens). the probability of getting a to the probability of getting a, then the probability of getting a. P(queen) = ; P(red) = ; P(red queen) = ( Example 3 The frequency distribution shows the volume of sales (in dollars) and the number of months a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between \$75,000 and \$124,999 next month? Sales Volume 0-24,999 25,000-49,999 50,000-74,999 75,000-99, , , , , , , , ,999 Months

8 Define Event A as monthly sales between and Define Event B as monthly sales between and Because events A and B are, the probability that the sales representative will sell between \$75,000 and \$124,999 next month is: P(A or B) = Example 4 A blood bank catalogs the types of blood, including positive or negative Rh-factor, given by donors during the last five days. The number of donors who gave each blood type is shown in the table. A donor is selected at random. 1. Find the probability that the donor has type O or type A blood. 2. Find the probability that the donor has type B or is Rh-negative. Blood Type Rh-Factor SOLUTIONS: 1. These events are. P(O or A) = P(O) + P(A) = O A B AB Total Positive Negative Total These events are. P(B or Rh neg) = P(B) + P(Rh neg) P(B and Rh neg) = Example 5 Use the graph to find the probability that a randomly selected draft pick is not a running back or a wide receiver. Define Events A and B Event A: Draft pick is a running back. Event B: Draft pick is a wide receiver. These events are, so the probability that the draft pick is a running pick or a wide receiver is: P(A or B) = P(A) + P(B) = To find the probability that the draft is NOT a running back or wide receiver, simply subtract the probability that he was one of those positions from 1 ( rule) = Section 3-4 Permutations A permutation is an arrangement of objects. The number of different permutations of n distinct object is n! The! Symbol means factorial and indicates that you start with the number given and multiply that times every number between that number and zero. 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880 The calculator will do factorials for you; Math Prb 4 accesses that feature. To find the number of ways that a permutation can occur, use the TI-84. Math PRB 2 Enter the total number of items in your list.

9 Press Math - PRB 2 and Enter Enter the number of items you wish to order and press enter. Distinguishable Permutations A distinguishable permutation must be done by hand. n! n 1!n 2!n 3! n k! Alpha and y= on your calculator will allow you to enter this as a fraction, making it easier. Again, the calculator will also let you enter the factorial symbol, saving time and reducing the chance of making a simple error in entering the numbers. Combinations A combination is a of r objects from a group of n objects regard to. To find the number of ways that a combination can occur, use the TI-84. Math PRB 3 Enter the total number of items in your list. Press Math - PRB 3 and Enter Enter the number of items you wish to select and press enter. Example 1 The objective of a 9 x 9 Sudoku number puzzle is to fill the grid so that each row, each column, and each 3 x 3 grid contains the digits 1 to 9, with no repeats. How many different ways can the first row of a blank 9 x 9 Sudoku grid be filled. There are digits that could fill the first spot, for the second spot, and so forth. This can be expressed as either, or as. 9! = 9P 9 also equals. So, there are ways to fill the first row of a 9 x 9 Sudoku number puzzle. Example 2 Find the number of ways of forming three-digit codes in which no digit is repeated. To form a three-digit code with no repeating digits, you need to select digits from a group of, so n = and r =. 10P 3 =. You could also simply say that you had choices for the first digit, choices for the second digit, and choices for the third digit. * * =. Example 3 Forty-three race cars started the 2007 Daytona 500. How many ways can the cars finish first, second, and third? You want to order 3 cars out of 43. This is a. = * * = cars could win the race. Once the winner has won, there are left to fight for 2 nd. Once 2 nd has been determined, there are left to battle it out for 3 rd. Example 4 A building contractor is planning to develop a sub-division. The sub-division is to consist of 6 one-story houses, 4 twostory houses and 2 split-level houses. In how many distinguishable ways can the houses be arranged? 12! 6!4!2! = different distinguishable ways to arrange the houses. Example 5 A state s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies? =. There are different combinations of 4 companies that can be selected from the 16 bidding companies.

10 Example 6 A student advisory board consists of 17 members. Three members serve as the board s chair, secretary, and webmaster. Each member is equally likely to serve any of the positions. What is the probability of selecting at random the three members that hold each position? There are different ways that the three positions can be filled. There are different ways that the three positions can be filled. The chance that you randomly pick the one correct outcome is Example 7 You have 11 letters consisting of one M, four I s, four S s and two P s. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word Mississippi? This is a distinguishable permutation question. There are 11! 1!4!4!2! different distinguishable ways to arrange those 11 letters. There are different distinguishable ways to arrange those 11 letters. 1 The probability of randomly picking the one arrangement that spells Mississippi is Example 8 Find the probability of being dealt five diamonds from a standard deck of playing cards. You need to determine how many ways you can get 5 diamonds, and then how many different 5 card hands are possible. Once you have these values, the number of ways to get 5 diamonds by the number of possible hands to find out how likely it is that you get 5 diamonds. 13C 5 = and 52C 5 = 1287 P(5 diamonds) =. 2,598,960 The probability of getting a diamond flush is approximately. Example 9 A food manufacturer is analyzing a sample of 400 corn kernels for the presence of a toxin. In this sample, three kernels have dangerously high levels of the toxin. If four kernels are randomly selected from the sample, what is the probability that exactly one kernel contains a dangerously high level of the toxin? Find the number of possible ways to choose one toxic kernel and three non-toxic kernels, those together and the answer by the number of ways to choose 4 kernels. This tells you how likely it is that you will randomly choose exactly one toxic kernel out of four. = and = 3 * 10,349,790 = There are different ways to select one toxic kernel out of 400. = There are different ways to select 4 kernels out of ,049,370 P(exactly 1 toxic kernel) =. 1,050,739,900 The probability of getting exactly one toxic kernel is approximately.

### 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

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

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

### Probability. 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

### 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

### Probability Concepts and Counting Rules

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

### Math 227 Elementary Statistics. Bluman 5 th edition

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

### 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

### 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

### Chapter 3: Probability (Part 1)

Chapter 3: Probability (Part 1) 3.1: Basic Concepts of Probability and Counting Types of Probability There are at least three different types of probability Subjective Probability is found through people

### Probability is the likelihood that an event will occur.

Section 3.1 Basic Concepts of is the likelihood that an event will occur. In Chapters 3 and 4, we will discuss basic concepts of probability and find the probability of a given event occurring. Our main

### 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)

### 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

### 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

### Probability - Chapter 4

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

### 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

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

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

### 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

### 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

### 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

### A Probability Work Sheet

A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we

### 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

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

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

### PROBABILITY. 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

### CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY

CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many real-world fields, such as insurance, medical research, law enforcement, and political science. Objectives:

### 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

### Raise your hand if you rode a bus within the past month. Record the number of raised hands.

166 CHAPTER 3 PROBABILITY TOPICS Raise your hand if you rode a bus within the past month. Record the number of raised hands. Raise your hand if you answered "yes" to BOTH of the first two questions. Record

### INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2

INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results

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

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

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

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

### 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.

### Chapter 11: Probability and Counting Techniques

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

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

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

### 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

### 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

### Statistics 1040 Summer 2009 Exam III

Statistics 1040 Summer 2009 Exam III 1. For the following basic probability questions. Give the RULE used in the appropriate blank (BEFORE the question), for each of the following situations, using one

### Date. Probability. Chapter

Date Probability Contests, lotteries, and games offer the chance to win just about anything. You can win a cup of coffee. Even better, you can win cars, houses, vacations, or millions of dollars. Games

### 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

### 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,

### Grade 6 Math Circles Fall Oct 14/15 Probability

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

### Math 141 Exam 3 Review with Key. 1. P(E)=0.5, P(F)=0.6 P(E F)=0.9 Find ) b) P( E F ) c) P( E F )

Math 141 Exam 3 Review with Key 1. P(E)=0.5, P(F)=0.6 P(E F)=0.9 Find C C C a) P( E F) ) b) P( E F ) c) P( E F ) 2. A fair coin is tossed times and the sequence of heads and tails is recorded. Find a)

### 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.

### Key Concepts. Theoretical Probability. Terminology. Lesson 11-1

Key Concepts Theoretical Probability Lesson - Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally

### 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

### 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

### Math 7 Notes - Unit 11 Probability

Math 7 Notes - Unit 11 Probability Probability Syllabus Objective: (7.2)The student will determine the theoretical probability of an event. Syllabus Objective: (7.4)The student will compare theoretical

### Mathematics 'A' level Module MS1: Statistics 1. Probability. The aims of this lesson are to enable you to. calculate and understand probability

Mathematics 'A' level Module MS1: Statistics 1 Lesson Three Aims The aims of this lesson are to enable you to calculate and understand probability apply the laws of probability in a variety of situations

### Chapter 4. Probability and Counting Rules. McGraw-Hill, Bluman, 7 th ed, Chapter 4

Chapter 4 Probability and Counting Rules McGraw-Hill, Bluman, 7 th ed, Chapter 4 Chapter 4 Overview Introduction 4-1 Sample Spaces and Probability 4-2 Addition Rules for Probability 4-3 Multiplication

### 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

### Outcomes: The outcomes of this experiment are yellow, blue, red and green.

(Adapted from http://www.mathgoodies.com/) 1. Sample Space The sample space of an experiment is the set of all possible outcomes of that experiment. The sum of the probabilities of the distinct outcomes

### 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

### 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

### Quiz 2 Review - on Notebook Paper Are You Ready For Your Last Quiz In Honors Math II??

Quiz 2 Review - on Notebook Paper Are You Ready For Your Last Quiz In Honors Math II?? Some things to Know, Memorize, AND Understand how to use are n What are the formulas? Pr ncr Fill in the notation

### NAME DATE PERIOD. Study Guide and Intervention

9-1 Section Title The probability of a simple event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. Outcomes occur at random if each outcome occurs by chance.

### AP Statistics Ch In-Class Practice (Probability)

AP Statistics Ch 14-15 In-Class Practice (Probability) #1a) A batter who had failed to get a hit in seven consecutive times at bat then hits a game-winning home run. When talking to reporters afterward,

### 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

### Additional Topics in Probability and Counting. Try It Yourself 1. The number of permutations of n distinct objects taken r at a time is

168 CHAPTER 3 PROBABILITY 3.4 Additional Topics in Probability and Counting WHAT YOU SHOULD LEARN How to find the number of ways a group of objects can be arranged in order How to find the number of ways

### 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

### Probability and Counting Rules. Chapter 3

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

### Unit 7 Central Tendency and Probability

Name: Block: 7.1 Central Tendency 7.2 Introduction to Probability 7.3 Independent Events 7.4 Dependent Events 7.1 Central Tendency A central tendency is a central or value in a data set. We will look at

### Chapter 3: Elements of Chance: Probability Methods

Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 3-4 2014-2015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,

### Unit 1 Day 1: Sample Spaces and Subsets. Define: Sample Space. Define: Intersection of two sets (A B) Define: Union of two sets (A B)

Unit 1 Day 1: Sample Spaces and Subsets Students will be able to (SWBAT) describe events as subsets of sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,

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

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

### 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,

### 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

### Probability as a general concept can be defined as the chance of an event occurring.

3. Probability In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. Probability as a general

### 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

### Section : Combinations and Permutations

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

### Intermediate Math Circles November 1, 2017 Probability I

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

### 4.3 Rules of Probability

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

### Probability. 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

### 5 Elementary Probability Theory

5 Elementary Probability Theory 5.1 What is Probability? The Basics We begin by defining some terms. Random Experiment: any activity with a random (unpredictable) result that can be measured. Trial: one

### Nwheatleyschaller s The Next Step...Conditional Probability

CK-12 FOUNDATION Nwheatleyschaller s The Next Step...Conditional Probability Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) Meery To access a customizable version of

### Basic Probability Concepts

6.1 Basic Probability Concepts How likely is rain tomorrow? What are the chances that you will pass your driving test on the first attempt? What are the odds that the flight will be on time when you go

### Independent and Mutually Exclusive Events

Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A

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

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

### Probability. Probabilty Impossibe Unlikely Equally Likely Likely Certain

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

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

Mathematical Ideas Chapter 2 Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) In one town, 2% of all voters are Democrats. If two voters

### The point value of each problem is in the left-hand margin. You must show your work to receive any credit, except on problems 1 & 2. Work neatly.

Introduction to Statistics Math 1040 Sample Exam II Chapters 5-7 4 Problem Pages 4 Formula/Table Pages Time Limit: 90 Minutes 1 No Scratch Paper Calculator Allowed: Scientific Name: The point value of

### 0-5 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins.

1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. d. a. Copy the table and add a column to show the experimental probability of the spinner landing on

### Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes

Worksheet 6 th Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of

### 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

### WEEK 7 REVIEW. Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.1)

WEEK 7 REVIEW Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.) Definition of Probability (7.2) WEEK 8-7.3, 7.4 and Test Review THE MULTIPLICATION

### MATH 2000 TEST PRACTICE 2

MATH 2000 TEST PRACTICE 2 1. Maggie watched 100 cars drive by her window and compiled the following data: Model Number Ford 23 Toyota 25 GM 18 Chrysler 17 Honda 17 What is the empirical probability that

### Chapter 4: Probability

Student Outcomes for this Chapter Section 4.1: Contingency Tables Students will be able to: Relate Venn diagrams and contingency tables Calculate percentages from a contingency table Calculate and empirical

### Probability Review before Quiz. Unit 6 Day 6 Probability

Probability Review before Quiz Unit 6 Day 6 Probability Warm-up: Day 6 1. A committee is to be formed consisting of 1 freshman, 1 sophomore, 2 juniors, and 2 seniors. How many ways can this committee be

### Chapter 6: Probability and Simulation. The study of randomness

Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes

### Applied Statistics I

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

### Statistics Intermediate Probability

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

### The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you \$5 that if you give me \$10, I ll give you \$20.)

The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you \$ that if you give me \$, I ll give you \$2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If

### Total. STAT/MATH 394 A - Autumn Quarter Midterm. Name: Student ID Number: Directions. Complete all questions.

STAT/MATH 9 A - Autumn Quarter 015 - Midterm Name: Student ID Number: Problem 1 5 Total Points Directions. Complete all questions. You may use a scientific calculator during this examination; graphing

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

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

### ECON 214 Elements of Statistics for Economists

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

### Chapter 4 Student Lecture Notes 4-1

Chapter 4 Student Lecture Notes 4-1 Basic Business Statistics (9 th Edition) Chapter 4 Basic Probability 2004 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic Probability Concepts Sample spaces and events,

### b) Find the exact probability of seeing both heads and tails in three tosses of a fair coin. (Theoretical Probability)

Math 1351 Activity 2(Chapter 11)(Due by EOC Mar. 26) Group # 1. A fair coin is tossed three times, and we would like to know the probability of getting both a heads and tails to occur. Here are the results

### 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.

### 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

### 3 The multiplication rule/miscellaneous counting problems

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

### 7 5 Compound Events. March 23, Alg2 7.5B Notes on Monday.notebook

7 5 Compound Events At a juice bottling factory, quality control technicians randomly select bottles and mark them pass or fail. The manager randomly selects the results of 50 tests and organizes the data