# PROBABILITY section. The Probability of an Event

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 4.3 Probability (4-3) PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques to find probabilities. The Probability of an Event The Addition Rule Complementary Events Odds The Probability of an Event In probability an experiment is a process such as tossing a coin, tossing a die, drawing a poker hand from a deck, or arranging people in a line. A sample space is the set of all possible outcomes to an experiment. An event is a subset of a sample space. For example, if we toss a coin, then the sample space consists of two equally likely outcomes, heads and tails. We write S H, T. The subset E H is the event of getting heads when the coin is tossed. We use n(s) to represent the number of equally likely outcomes in the sample space S and n(e ) to represent the number of outcomes in the event E. For the example of tossing a coin, n(s) 2 and n(e). The Probability of an Event If S is a sample space of equally likely outcomes to an experiment and the event E is a subset of S, then the probability of E, P(E), is defined to be n(e) P(E). n(s) When S H, T and E H, n(e) P(E) 2. n(s) So the probability of getting heads on a single toss of a coin is. 2 If E is the event of getting 2 heads on a single toss of a coin, then n(e) 0 and P(E) If E is the event of getting fewer than 2 heads on a single toss of a coin, then for either outcome H or T we have fewer than 2 heads. So E H, T, n(e) 2, and P(E) 2 2. Note that the probability of an event is a number between 0 and inclusive, being the probability of an event that is certain to occur and 0 being the probability of an event that is impossible to occur. E X A M P L E Rolling a die What is the probability of getting a number larger than 4 when a single die is rolled? When we roll a die, we count the number of dots showing on the upper face of the die. So the sample space of equally likely outcomes is S, 2, 3, 4, 5, 6. Since only 5 and 6 are larger than 4, E 5, 6. According to the definition of probability, n(e) P(E) n(s)

2 728 (4-4) Chapter 4 Counting and Probability M A T H A T W O R K The probability experiments discussed in this chapter are not just textbook examples that have no relationship to real life. For example, if a couple plans to have 6 children and the probability of having a girl on each try is 2, then the couple can expect to have 3 girls. If you guess at the answer to each question of a 00-question, 5-choice multiple-choice test, then you have 5 probability of getting each question correct, and you can expect to get 20 questions correct. Try it. The expected number of successes is the product of the probability of success and the number of tries. Lotteries provide us an opportunity to observe massive probability experiments. In the Florida Lottery you can win by choosing 6 numbers from the numbers through 49 and matching the 6 numbers chosen by the Florida Lottery. There are C(49, 6) ways to choose 6 numbers from 49, so the probability of winning on any individual try is. C(49, 6) 3,98 3,86 In the fall of 990 the weekly drawing frequently had relatively few participants, and consequently there was no winner for many weeks. When the prize got up to \$06.5 million, the lottery got national attention. People came from everywhere to participate. During the week prior to September 5, 990, 09,63,978 tickets were sold. We expected 3,98 09,63, winners. On September 5 the 3,86 winning numbers were announced, and 6 winners shared the prize. Of course, probability cannot predict the future like a fortune-teller, but the power of probability to tell us what to expect is truly amazing. LOTTERIES E X A M P L E 2 Tossing coins What is the probability of getting at least one head when a pair of coins is tossed? Since there are 2 equally likely outcomes for the first coin and 2 equally likely outcomes for the second coin, by the fundamental counting principle there are 4 equally likely outcomes to the experiment of tossing a pair of coins. We can list the outcomes as ordered pairs: S (H, H), (H, T), (T, H), (T, T). Since 3 of these outcomes result in at least one head, E (H, H), (H, T), (T, H), and n(e) 3. So n(e) P(E) 3 4. n(s) E X A M P L E 3 Rolling a pair of dice What is the probability of getting a sum of 6 when a pair of dice is rolled?

3 Since there are 6 equally likely outcomes for each die, there are equally likely outcomes to the experiment of rolling the pair. We can list the outcomes as ordered pairs: S (, ), (, 2), (, 3), (, 4), (, 5), (, 6), (2, ), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, ), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, ), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, ), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, ), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) 4.3 Probability (4-5) 729 The sum of the numbers is 6, describes the event E (5, ), (4, 2), (3, 3), (2, 4), (, 5). So n(e) 5 P(E). n(s) The Addition Rule In tossing a pair of dice, let A be the event that doubles occurs and B be the event that the sum is 4. We can list the following events and their probabilities: 6 A (, ), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) and P(A) 3 B (3, ), (2, 2), (, 3) and P(B) A B (, ), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (3, ), (, 3) and P(A B) 3 86 A B (2, 2) and P(A B) 8 Note that the probability of doubles or a sum of 4, P(A B), is and This equation makes sense because there is one outcome, (2, 2), that is in both the events A and B. This example illustrates the addition rule. The Addition Rule If A and B are any events in a sample space, then P(A B) P(A) P(B) P(A B). If P(A B) 0, then A and B are called mutually exclusive events and P(A B) P(A) P(B). Note that for mutually exclusive events it is impossible for both events to occur. The addition rule for mutually exclusive events is a special case of the general addition rule.

4 730 (4-6) Chapter 4 Counting and Probability E X A M P L E 4 E X A M P L E 5 The addition rule At Downtown College 60% of the students are commuters (C), 50% are female (F), and 30% are female commuters. If a student is selected at random, what is the probability that the student is either a female or a commuter? By the addition rule the probability of selecting either a female or a commuter is P(F C) P(F) P(C) P(F C) The addition rule with dice In rolling a pair of dice, what is the probability that the sum is 2 or at least one die shows a 2? Let A be the event that the sum is 2 and B be the event that at least one die shows a 2. Since A occurs on only one of the equally likely outcomes (see Example 3), P(A). Since B occurs on of the equally likely outcomes, P(B). Since A and B are mutually exclusive, we have P(A B) P(A) P(B) 2 3. Complementary Events If the probability of rain today is 60%, then the probability that it does not rain is 40%. Rain and not rain are called complementary events. There is no possibility that both occur, and one of them must occur. If A is an event, then A (read A bar or A complement ) represents the complement of the event A. Note that complementary events are mutually exclusive, but mutually exclusive events are not necessarily complementary. Complementary Events Two events A and A are called complementary events if A A and P(A) P(A). E X A M P L E 6 E X A M P L E 7 Complementary events What is the probability of getting a number less than or equal to 4 when rolling a single die? We saw in Example that getting a number larger than 4 when rolling a single die has probability 3. The complement to getting a number larger than 4 is getting a number less than or equal to 4. So the probability of getting a number less than or equal to 4 is 2 3. Complementary events If the probability that White Lightning will win the Kentucky Derby is 0.5, then what is the probability that White Lightning does not win the Kentucky Derby?

5 4.3 Probability (4-7) 73 study tip Study for the final exam by working actual test questions. Be sure to rework all of your tests. Do the chapter tests in this book. You can get more tests to work by asking students or instructors for tests that were given in other classes of this course. Let W be winning the Kentucky Derby and N be not winning the Kentucky Derby. Since W and N are complementary events, we have P(W) P(N). So P(N) P(W) Odds If the probability is 2 3 that the Giants win the Super Bowl and 3 that they lose, then they are twice as likely to win as they are to lose. We say that the odds in favor of the Giants winning the Super Bowl are 2 to. Notice that odds are not probabilities. Odds are ratios of probabilities. We usually write odds as ratios of whole numbers. Odds If A is any event, then the odds in favor of A is the ratio P(A) to P(A) and the odds against A is the ratio of P(A) to P(A). E X A M P L E 8 Determining odds What are the odds in favor of getting a sum of 6 when rolling a pair of dice? What are the odds against a sum of 6? 5 In Example 3 we found the probability of a sum of 6 to be. So the probability of the complement (the sum is not 6) is 3. The odds in favor of getting a sum of 6 are 5 to 3. Multiply each fraction by to get the odds 5 to 3. The odds against a sum of 6 are 3 to 5. E X A M P L E 9 helpful hint Odds and probability are often confused, even by people who write lottery tickets. If the probability of winning a lottery is, then the probability of losing is, and the 00 odds in favor of winning are to 99. Many lottery tickets will state (incorrectly) that the odds in favor of winning are to 00. Determining probability given the odds If the odds in favor of Daddy s Darling winning the third race at Delta Downs are 4 to, then what is the probability that Daddy s Darling wins the third race? Since 4 to is the ratio of the probability of winning to not winning, the probability of winning is four times as large as the probability of not winning. Let P(W) x and P(W ) 4x. Since P(W ) P(W ), we have 4x x, or 5x, or x 5. So the probability of winning is 4 5. We can write the idea found in Example 9 as a strategy for converting from odds to probabilities. Strategy for Converting from Odds to Probability If the odds in favor of event E are a to b, then a b P(E) and P(E). a b a b

7 4.3 Probability (4-9) If a single coin is tossed twice, then what is the probability of getting a) heads followed by tails? c) a tail on the second toss? b) two heads in a row? d) exactly one tail? 4, 4, 2, 2. If a pair of dice is tossed, then what is the probability of getting a) a pair of 2 s? d) a sum greater than? b) at least one 2? e) a sum less than 2? c) a sum of 7?, 6,,,0 2. If a single die is tossed twice, then what is the probability of getting a) a followed by a 2? b) a sum of 3? 8 c) a 6 on the second toss? 6 d) no more than two 5 s? e) an even number followed by an odd number? 4 3. A ball is selected at random from a jar containing 3 red balls, 4 yellow balls, and 5 green balls. What is the probability that a) the ball is red? 4 b) the ball is not yellow? 2 3 c) the ball is either red or green? 2 3 d) the ball is neither red nor green? 3 FIGURE FOR EXERCISE 3 4. A committee consists of Democrat, 5 Republicans, and 6 independents. If one person is randomly selected from the committee to be the chairperson, then what is the probability that a) the person is a Democrat? 2 b) the person is either a Democrat or a Republican? 2 7 c) the person is not a Republican? 2 5. A jar contains 0 balls numbered through 0. Two balls are randomly selected one at a time without replacement. What is the probability that a) is selected first and 2 is selected second? 9 0 b) the sum of the numbers selected is 3? c) the sum of the numbers selected is 6? A small company consists of a president, a vice-president, and 4 salespeople. If 2 of the 6 people are randomly selected to win a Hawaiian vacation, then what is the probability that none of the salespeople is a winner? If a 5-card poker hand is drawn from a deck of 52, then what is the probability that a) the hand contains the ace, king, queen, jack, and ten of spades? 2,59 8,960 b) the hand contains one 2, one 3, one 4, one 5, and one 6? 024 2,598, If 5 people with different names and different weights randomly line up to buy concert tickets, then what is the probability that a) they line up in alphabetical order? 20 b) they line up in order of increasing weight? 20 Use the addition rule to solve each problem. See Examples 4 and Among the drivers insured by American Insurance, 65% are women, 38% of the drivers are in a high-risk category, and 24% of the drivers are high-risk women. If a driver is randomly selected from that company, what is the probability that the driver is either high-risk or a woman? What is the probability of getting either a sum of 7 or at 5 least one 4 in the toss of a pair of dice? 2 2. A couple plans to have 3 children. Assuming males and females are equally likely, what is the probability that they have either 3 boys or 3 girls What is the probability of getting a sum of 0 or a sum of 5 7 in the toss of a pair of dice? 23. What is the probability of getting either a heart or an ace 4 when drawing a single card from a deck of 52 cards? What is the probability of getting either a heart or a spade when drawing a single card from a deck of 52 cards? 2 Solve each problem. See Examples 6 and If the probability of surviving a head-on car accident at 55 mph is 0.005, then what is the probability of not surviving? If the probability of a tax return not being audited by the IRS is 0.97, then what is the probability of a tax return being audited? A pair of dice is tossed. What is the probability of a) getting a pair of 4 s? b) not getting a pair of 4 s? 3 5 c) getting at least one number that is not a 4? Three coins are tossed. What is the probability of a) getting three heads? 8 b) not getting three heads? 7 8 c) getting at least one tail? 7 8

8 734 (4-20) Chapter 4 Counting and Probability Solve each problem. See Examples 8 and If the probability is 60% that the eye of Hurricane Edna comes ashore within 30 miles of Charleston, then what are the odds in favor of the eye of Edna coming ashore within 30 miles of Charleston? 3 to 2 FIGURE FOR EXERCISE 38 FIGURE FOR EXERCISE If the probability that a Sidewinder missile hits its target is 8 9, then what are the odds a) in favor of the Sidewinder hitting its target? 8 to b) against the Sidewinder hitting its target? to 8 3. If the probability that the stock market goes up tomorrow is 3 5, then what are the odds a) in favor of the stock market going up tomorrow? 3 to 2 b) against the stock market going up tomorrow? 2 to If the probability of a coal miners strike this year is 0, then what are the odds a) in favor of a strike? 9 to b) against a strike? to If the odds are 3 to in favor of the Black Hawks winning their next game, then a) what are the odds against the Black Hawks winning their next game? to 3 b) what is the probability that the Black Hawks win their next game? If the odds are 5 to against the Democratic presidential nominee winning the election, then a) what are the odds in favor of the Democrat winning the election? to 5 b) what is the probability that the Democrat wins the election? What are the odds in favor of getting exactly 2 heads in 3 tosses of a coin? 3 to 5. What are the odds in favor of getting a 6 in a single toss of a die? to What are the odds in favor of getting a sum of 8 when tossing a pair of dice? 5 to What are the odds in favor of getting at least one 6 when tossing a pair of dice? to If one million lottery tickets are sold and only one of them is the winning ticket, then what are the odds in favor of winning if you hold a single ticket? to 999, What are the odds in favor of winning a lottery where you must choose 6 numbers from the numbers through 49? to 3,983,85 4. If the odds in favor of getting 5 heads in 5 tosses of a coin are to 3, then what is the probability of getting 5 heads in 5 tosses of a coin? If the odds against Smith winning the election are 2 to 5, then what is the probability that Smith wins the election? 5 7 GETTING MORE INVOLVED 43. In the Louisiana Lottery a player chooses 6 numbers from the numbers through 44. You win the big prize if the 6 chosen numbers match the 6 winning numbers chosen on Saturday night. a) What is the probability that you choose all 6 winning numbers? b) What is the probability that you do not get all 6 winning numbers? c) What are the odds in favor of winning the big prize with a single entry?, 7, 059, 05, to 7,059,05 7,059,052 7, 059, In the Louisiana Power Ball a player chooses 5 numbers from the numbers through 49 and one number (the power ball) from through 42. a) How many ways are there to choose the 5 numbers and, choose the power ball? b) What is the probability of winning the big prize in the Power Ball Lottery? c) What are the odds in favor of winning the big prize? 80,089,28,, to 80,089,27 80,08 9,28

### 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

### 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

### 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,

### 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

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

Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called

### 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

Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,

### 36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### number of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.

12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.

### Understanding. Probability and Long-Term Expectations. Chapter 16. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Understanding Chapter 16 Probability and Long-Term Expectations Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Thought Question 1: Two very different queries about probability: a. If

### 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

### (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

### 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

### 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

### 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

### 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.

### 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

### 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

### Lesson 1: Experimental and Theoretical Probability

Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.

### Probability And Odds Examples

Probability And Odds Examples. Will the Cubs or the Giants be more likely to win the game? What is the chance of drawing an ace from a deck of cards? What are the possibilities of rain today? What are

### Probability definitions

Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating

### Probability OPRE 6301

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

### Math 118 Study Guide. This study guide is for practice only. The actual question on the final exam may be different.

Math 118 Study Guide This study guide is for practice only. The actual question on the final exam may be different. Convert the symbolic compound statement into words. 1) p represents the statement "It's

### The number of phone calls to the attendance office of a high school on any given school day A) continuous B) discrete

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) State whether the variable is discrete or continuous.

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

### MAT 1000. Mathematics in Today's World

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

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

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

### 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

### 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

### 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

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2

Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability

### Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

### Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

### Math 150 Sample Exam #2

Problem 1. (16 points) TRUE or FALSE. a. 3 die are rolled, there are 1 possible outcomes. b. If two events are complementary, then they are mutually exclusive events. c. If A and B are two independent

### 2. Three dice are tossed. Find the probability of a) a sum of 4; or b) a sum greater than 4 (may use complement)

Probability Homework Section P4 1. A two-person committee is chosen at random from a group of four men and three women. Find the probability that the committee contains at least one man. 2. Three dice

### Conditional Probability and General Multiplication Rule

Conditional Probability and General Multiplication Rule Objectives: - Identify Independent and dependent events - Find Probability of independent events - Find Probability of dependent events - Find Conditional

### 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)

### 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

### Homework 8 Solutions

CSE 21 - Winter 2014 Homework Homework 8 Solutions 1 Of 330 male and 270 female employees at the Flagstaff Mall, 210 of the men and 180 of the women are on flex-time (flexible working hours). Given that

### Chapter 7 Probability. Example of a random circumstance. Random Circumstance. What does probability mean?? Goals in this chapter

Homework (due Wed, Oct 27) Chapter 7: #17, 27, 28 Announcements: Midterm exams keys on web. (For a few hours the answer to MC#1 was incorrect on Version A.) No grade disputes now. Will have a chance to

### Probability and Venn diagrams UNCORRECTED PAGE PROOFS

Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve

### Probability (Day 1 and 2) Blue Problems. Independent Events

Probability (Day 1 and ) Blue Problems Independent Events 1. There are blue chips and yellow chips in a bag. One chip is drawn from the bag. The chip is placed back into the bag. A second chips is then

### 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

### Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS

Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS 1. Give two examples of ways that we speak about probability in our every day lives. NY REASONABLE ANSWER OK. EXAMPLES: 1) WHAT

### Decision Making Under Uncertainty. Professor Peter Cramton Economics 300

Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate

### 4.4 Conditional Probability

4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

### 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.

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Probability and Expected Value

Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

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

Ch. 4 Discrete Probability Distributions 4.1 Probability Distributions 1 Decide if a Random Variable is Discrete or Continuous 1) State whether the variable is discrete or continuous. The number of cups

### Counting principle, permutations, combinations, probabilities

Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing

### 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

### Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR.

Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR. 1. Urn A contains 6 white marbles and 4 red marbles. Urn B contains 3 red marbles and two white

### CHAPTER 3: PROBABILITY TOPICS

CHAPTER 3: PROBABILITY TOPICS Exercise 1. In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities

### Chapter 4 Probability

The Big Picture of Statistics Chapter 4 Probability Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time) 2 What is probability?

### 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..

### A Few Basics of Probability

A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

### 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

### 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

### PROBABILITY. Chapter Overview

Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability

### 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

### CHAPTER 4: DISCRETE RANDOM VARIABLE

CHAPTER 4: DISCRETE RANDOM VARIABLE Exercise 1. A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following

### 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

### Mathematical goals. Starting points. Materials required. Time needed

Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

### 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,

### Review for Test 2. Chapters 4, 5 and 6

Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair six-sided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

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

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

### HANDLING DATA. Use the vocabulary of probability

HANDLING DATA Pupils should be taught to: Use the vocabulary of probability As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: fair, unfair, likely, unlikely, equally

### 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

### 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.

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

Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.

### 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

### 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

### 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

### Solutions to Self-Help Exercises 1.3. b. E F = /0. d. G c = {(3,4),(4,3),(4,4)}

1.4 Basics of Probability 37 Solutions to Self-Help Exercises 1.3 1. Consider the outcomes as ordered pairs, with the number on the bottom of the red one the first number and the number on the bottom of

### 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

### Stats Review Chapters 5-6

Stats Review Chapters 5-6 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

### 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

### 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

### Probability and Counting

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

### The Casino Lab STATION 1: CRAPS

The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

### Chapter 4. Chapter 4. Probability. Probability

Chapter 4 Probability 1 4-1 Overview Chapter 4 Probability 4-2 Fundamentals 4-3 Addition Rule 4-4 Multiplication Rule: Basics 2 Objectives Overview develop sound understanding of probability values used

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### AP Stats Fall Final Review Ch. 5, 6

AP Stats Fall Final Review 2015 - Ch. 5, 6 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

### Worked examples Basic Concepts of Probability Theory

Worked examples Basic Concepts of Probability Theory Example 1 A regular tetrahedron is a body that has four faces and, if is tossed, the probability that it lands on any face is 1/4. Suppose that one

### 4.3. Addition and Multiplication Laws of Probability. Introduction. Prerequisites. Learning Outcomes. Learning Style

Addition and Multiplication Laws of Probability 4.3 Introduction When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring

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

Ch. - Problems to look at 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

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant

### PROBABILITY. SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and 1.

PROBABILITY SIMPLE PROBABILITY SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and. There are two categories of simple probabilities. THEORETICAL

### Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance

### 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. Find the mean for the given sample data. 1) Bill kept track of the number of hours he spent

### NAME: NUMERACY GAMES WITH DICE AND CARDS. A South African Numeracy Chair initiative

NAME: NUMERACY GAMES WITH DICE AND CARDS A South African Numeracy Chair initiative CONTENTS Games in the classroom 3 Benefits 3 Mathematical games are 'activities' which: 3 More benefits 3 Competition

### 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