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

Size: px
Start display at page:

Transcription

1 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 of winning the game, what is the sample space they must use? The sample space is the set of all possible outcomes. Therefore S = [1, 2, 3, 4, 5, 6]. 2. A board game requires each player to draw a card. Each card has a different number on it. The player with the highest number wins. Is the probability of a player winning an independent or dependent event? Why? The event is dependent because the probability that a winning card will be drawn is changed every time a card is drawn. Therefore, the results of the other players draws affect the probability that a particular player will win. 3. A box contains 12 red pens, 14 blue pens, and 6 black pens. If a red pen is drawn first and kept out of the bag, what fraction represents the probability that a black pen will be drawn second? 6/31. There are 6 black pens left and the red pen was removed, so the original sample size of 32 is reduced to A die has been rolled seven times in a row. Each time, the die landed on six. What is the probability that the die will land on six on the eighth roll? Why? The probability of landing on six is 1/6 because there are six sides to the die. The roll is an independent event, so the results of the previous rolls have no effect on the outcome of the eighth roll. 5. The probability that it will rain today is 70%. The probability that it will just be cloudy is 20%, and the probability that it will be sunny and clear is 10%. Is today s weather a biased event? Yes, the event is biased because the outcomes have different probabilities. 6. What is the probability that it will rain or that it will be sunny and clear? These are mutually exclusive events, so we simply add the probabilities together. P= =.8 = 80%.

2 7. If it rains, a gardener has a 40% chance of having a day off on the following day. If it doesn t rain, he has to work the next day. What is the probability that the gardener will have to work tomorrow? P(day off tomorrow) = P(rain)(1-P(day off)) + P(no rain) = (.7)(.6) +.3 = =.72 = 72%. 8. What is the probability that each spinner will land on purple? P(both on purple) = P(first spinner on purple) x P(second spinner on purple) = 2/8 x 2/8 = 4/64 = 1/16.

3 Use the following tree to answer questions What is the value of X? The value of X is 1-.6 = What is the probability that Event D will be the outcome? The probability of event D is equal to.8 x.1 = If A represents that it will snow and there will be a snow day, B represents that it it will snow and there won t be a snow day, C represents that it won t snow and there won t be a snow day, and D represents that it won t snow and there will be a snow day, what is the probability that there won t be a snow day? P(no snow day) = P(B AND C) = (.2)(.6) + (.8)(.9) = =.84 = 84%. 12. What is the probability that there will be a snow day? P(snow day) = P(A AND C) = (.2)(.4) + (.8)(.1) = =.16 = 16% (which is also 1 P(no snow day))

4 13. What probability model should be used when examining simple events? A systematic list can be used to model simple events effectively. 14. A game involves drawing 16 red coins, 12 blue coins and 3 green coins from a bag. A player wins if they draw a blue or green coin and loses if they draw a red coin. Is this game fair? Explain your answer. The game is not fair because the probability of winning is P(blue coin) + P(green coin) = 15/31, whereas the probability of losing is P(red coin) = 16/31. Because it is more likely that a player will lose the game, the game cannot be considered fair. 15. A game involves drawing 16 red coins, 13 blue coins and 3 green coins from a bag. A player wins if they draw a blue or green coin and loses if they draw a red coin. Is this game fair? Explain your answer. The game is fair because the probability of winning is P(blue coin) + P(green coin) = 16/32 = 1/2, whereas the probability of losing is P(red coin) = 16/32 = 1/2. Because the probability of losing the game is the same as the probability of winning the game, the game is considered fair. 16. What model should be used when performing complex probability calculations with multiple AND functions? Explain your answer. A tree diagram should be used. The tree diagram provides easy visual cues for and calculations. All one has to do is follow a path from the root of the tree to a node, multiplying the probabilities on the branches along the way. 17. At a given company, an employee is 42% likely to be male and 58% likely to be a female. An employee is also 30% likely to be an engineer, 50% likely to be a manager, 25% likely to be a researcher and 10% likely to be a technical writer. If Event A = male manager or male engineer, what is event A? Event A = male researcher or male tech writer or female. These are all the possible outcomes that aren t a male manager or a male engineer, and therefore form the complement of A. 18. At a given company, an employee is 42% likely to be male and 58% likely to be a female. An employee is also 30% likely to be an engineer, 50% likely to be a manager, 25% likely to be a researcher and 10% likely to be a technical writer. What is the probability of the union of being an engineer or a technical writer? P(E U TW) = P(E) + P(TW) P(E)P(TW) = (.3)(.1) = =.37 = 37%

5 19. Given Event A, Event B, and Event C. Event A and Event B are mutually exclusive. Event A and Event C are not mutually exclusive. P(A) = 0.45 P(B) = 0.35 P(C) = 0.25 What is the probability of the complement of Event B? P(B) + P(B ) = 1. Therefore, P(B ) = =.65 = 65% 20. Given Event A, Event B, and Event C. Event A and Event B are mutually exclusive. Event A and Event C are not mutually exclusive. P(A) = 0.55 P(B) = 0.35 P(C) = 0.66 What is the probability of the complement of Event A C? P(A C) = P(A)P(C) =.55 x.66 =.363. Thus, the probability of the complement is = What is the probability that a person will flip a coin three times and that it will land heads up three times in a row? We know that the probability of the coin coming up heads once is.5. So, we multiply.5 x.5 x.5 = A friend offers to play a game where you pay him \$2 if the roll of a 6-sided die comes up at 1, 2, 3, or 4, and he pays you \$3 if the die comes up a 5 or 6. What is the expected value of a round for you if you play the game? We find the probability of each possible outcome, multiply it by the dollar amount, and then add it together to get the expected value. So, we have (4 x (-2 x 1/6)) + (3 x (3 x 1/6)) = - 1/3.

6 23. Below is a spinner. What is the probability of landing on a number divisible by 2? We divide the number of desired events by the number of possible outcomes to get 4/8 = Using the spinner below, what is the probability of landing on a number divisible by 2 twice in a row? We divide the number of desired events by the number of possible outcomes to get.5 for landing on a number divisible by 2 once. Then, we multiply.5 x.5 =.25 to get the probability of landing on a number divisible by 2 twice. 25. If there is an event in which three decisions must be made and there are four choices available for each decision, how many potential outcomes are there? By the Fundamental Principle of Counting, we multiply the number of choices available to each decision together: 4 x 4 x 4 = 64 possible outcomes

7 26. How many potential combinations can be made from a lottery ticket with four digits, the first two digits being any letter from the alphabet and the last two being any number 0-9. By the Fundamental Principle of Counting, we have 26 x 26 x 10 x 10 = 67,600 possible outcomes. 27. Explain the difference between dependent and independent events. In an independent event, the choice made for a particular decision does not affect the choices available to any other decision. If the event is dependent, then (for at least one decision) making a decision affects the choices available to (at least) one other decision. 28. If we remove all the kings and queens from a 52 card deck, how many unique four card sequences can we draw? First, subtract 8 from 52 to account for the missing kings and queens. Then, using the Fundamental Principle of Counting, we have the potential number of outcomes equal to 44 x 43 x 42 x 41 = 3,258,024 possible outcomes. 29. What classifies a permutation? In a permutation, the order of the elements matters. Therefore, a different ordering of the same elements counts as a different permutation. 30. How many three digit permutations can be made from the numbers 1, 2, 3, 4, and 5 if repetition is not allowed? Since repetition isn t allowed, we use the equation n! / (n-r)! = 5!/(5-3)! = 5!/2! = 60 permutations. 31. How many three digit permutations can be made from the numbers 1, 2, 3, 4, and 5 if repetition is allowed? Since repetition is allowed, we use the equation n r = 3 5 = 243 permutations

8 32. How many possible codes can be formed with a locker that asks for four numbers, numbers cannot be used more than once, and any number can be 0-9? This locker code does not allow repetition. Therefore, we use n! / (n-r)! = 10! / (10-4)! = 10! / 6! = 5,040 permutations. 33. What classifies a combination? In combinations, the order of the elements doesn t matter. Therefore, different orderings of the same elements are considered the same combination. 34. How many combinations without repetition are possible if we have four initial choices for three decisions? This is a combination without repetition, so we use n! / r!(n-r)! = 4! / (3!)(4-3)! = 4! / (3!)(1!) = 4 combinations. 35. How many combinations with repetition are possible if we have four initial choices for three decisions? This is a combination with repetition, so we use (n+r-1)! / r! (n-1)! = 6! / (3!)(3!) = 20 combinations. 36. If Becca is at the store and can buy any three fruit (the store sells apples, oranges, pears, bananas, and kiwis), how many combinations of fruit can she choose? This is an example of a combination with repetition allowed. We use (n+r-1) / r! (n-1!) = 7! / (3!)(4!) = 35 combinations.

9 37. Given the same number of elements and choices to make, which will there be more of: permutations or combinations? Why? There will be more permutations. Because order matters in permutations, there are naturally more permutations given the same number of elements. This is easily identified in the equation for combinations without repetition, which is exactly the same as the one for permutations except we divide by r! 38. If we want to make a five letter word using any letters in the alphabet, how many possible words can we make (a word counts as any sequence of letters in this case)? Feel free to leave your answer in exponential or factorial form. This is a permutation because the order of the letters matters. Repetition is allowed because we can use any letter in the alphabet. Therefore, P(n,r) = n r = 26 5 combinations. 39. How many sandwiches can we make with one slice of turkey, one slice of ham, one slice of cheese, one dollop of mayonnaise, one dollop of ketchup, and one dollop of mustard if we are only using four ingredients? This is a combination because the order of the ingredients doesn t matter. Repetition is not allowed because we have a limited supply of each ingredient. Therefore, C(n,r) = n! / r! (n-r)! = 6! / (4!)(2!) = 15 combinations. 40. How many sandwiches can we make with access to unlimited turkey, ham, cheese, mayonnaise, ketchup, and mustard if we are only using four ingredients? This is a combination because the order of the ingredients doesn t matter. Repetition is allowed because we have an unlimited supply of each ingredient. Therefore, C(n,r) = (n+r-1)! / r! (n-1)! = 9! / (4!)(5!) = 126 combinations.

### 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

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

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

### 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 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 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.

### Lesson Lesson 3.7 ~ Theoretical Probability

Theoretical Probability Lesson.7 EXPLORE! sum of two number cubes Step : Copy and complete the chart below. It shows the possible outcomes of one number cube across the top, and a second down the left

### 10-4 Theoretical Probability

Problem of the Day A spinner is divided into 4 different colored sections. It is designed so that the probability of spinning red is twice the probability of spinning green, the probability of spinning

### Find the probability of an event by using the definition of probability

LESSON 10-1 Probability Lesson Objectives Find the probability of an event by using the definition of probability Vocabulary experiment (p. 522) trial (p. 522) outcome (p. 522) sample space (p. 522) event

### 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

### 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:

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

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

### EECS 203 Spring 2016 Lecture 15 Page 1 of 6

EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including

### 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.

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

### April 10, ex) Draw a tree diagram of this situation.

April 10, 2014 12-1 Fundamental Counting Principle & Multiplying Probabilities 1. Outcome - the result of a single trial. 2. Sample Space - the set of all possible outcomes 3. Independent Events - when

### 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

### Applications. 28 How Likely Is It? P(green) = 7 P(yellow) = 7 P(red) = 7. P(green) = 7 P(purple) = 7 P(orange) = 7 P(yellow) = 7

Applications. A bucket contains one green block, one red block, and two yellow blocks. You choose one block from the bucket. a. Find the theoretical probability that you will choose each color. P(green)

### Chapter 2. Permutations and Combinations

2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find

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

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

### 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 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

### Algebra 1B notes and problems May 14, 2009 Independent events page 1

May 14, 009 Independent events page 1 Independent events In the last lesson we were finding the probability that a 1st event happens and a nd event happens by multiplying two probabilities For all the

### 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

### 10. Because the order of selection doesn t matter: selecting 3, then 5 is the same as selecting 5, then 3. 25! 24 = 300

Chapter 6 Answers Lesson 6.1 1. li, lo, ln, ls, il, io, in, is, ol, oi, on, os, nl, ni, no, ns, sl, si, so, sn 2. 5, 4, 5 4 = 20, 6 5 = 30 3. (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9) (2,3) (2,4)

### 2. A bubble-gum machine contains 25 gumballs. There are 12 green, 6 purple, 2 orange, and 5 yellow gumballs.

A C E Applications Connections Extensions Applications. A bucket contains one green block, one red block, and two yellow blocks. You choose one block from the bucket. a. Find the theoretical probability

### , x {1, 2, k}, where k > 0. (a) Write down P(X = 2). (1) (b) Show that k = 3. (4) Find E(X). (2) (Total 7 marks)

1. The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). (1) Show that k = 3. Find E(X). (Total 7 marks) 2. In a game

### Probability Unit 6 Day 3

Probability Unit 6 Day 3 Warm-up: 1. If you have a standard deck of cards in how many different hands exists of: (Show work by hand but no need to write out the full factorial!) a) 5 cards b) 2 cards 2.

### , -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,

### Probability Name: To know how to calculate the probability of an outcome not taking place.

Probability Name: Objectives: To know how to calculate the probability of an outcome not taking place. To be able to list all possible outcomes of two or more events in a systematic manner. Starter 1)

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

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

### (a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?

Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent

### This unit will help you work out probability and use experimental probability and frequency trees. Key points

Get started Probability This unit will help you work out probability and use experimental probability and frequency trees. AO Fluency check There are 0 marbles in a bag. 9 of the marbles are red, 7 are

### 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

### PRE TEST. Math in a Cultural Context*

P grade PRE TEST Salmon Fishing: Investigations into A 6P th module in the Math in a Cultural Context* UNIVERSITY OF ALASKA FAIRBANKS Student Name: Grade: Teacher: School: Location of School: Date: *This

### Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1

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

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

### Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)

12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the

### 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

### 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,

### 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)

### 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

### 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

### Adriana tosses a number cube with faces numbered 1 through 6 and spins the spinner shown below at the same time.

Domain 5 Lesson 9 Compound Events Common Core Standards: 7.SP.8.a, 7.SP.8.b, 7.SP.8.c Getting the Idea A compound event is a combination of two or more events. Compound events can be dependent or independent.

### 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

### 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

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

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

### 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

### Foundations to Algebra In Class: Investigating Probability

Foundations to Algebra In Class: Investigating Probability Name Date How can I use probability to make predictions? Have you ever tried to predict which football team will win a big game? If so, you probably

### CISC 1400 Discrete Structures

CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Mega-million Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest

### 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

### Revision Topic 17: Probability Estimating probabilities: Relative frequency

Revision Topic 17: Probability Estimating probabilities: Relative frequency Probabilities can be estimated from experiments. The relative frequency is found using the formula: number of times event occurs.

### 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

### 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

### Compound Probability. Set Theory. Basic Definitions

Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic

### b. 2 ; the probability of choosing a white d. P(white) 25, or a a. Since the probability of choosing a

Applications. a. P(green) =, P(yellow) = 2, or 2, P(red) = 2 ; three of the four blocks are not red. d. 2. a. P(green) = 2 25, P(purple) = 6 25, P(orange) = 2 25, P(yellow) = 5 25, or 5 2 6 2 5 25 25 25

### Honors Precalculus Chapter 9 Summary Basic Combinatorics

Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each

### Tail. Tail. Head. Tail. Head. Head. Tree diagrams (foundation) 2 nd throw. 1 st throw. P (tail and tail) = P (head and tail) or a tail.

When you flip a coin, you might either get a head or a tail. The probability of getting a tail is one chance out of the two possible outcomes. So P (tail) = Complete the tree diagram showing the coin being

### PRE TEST KEY. Math in a Cultural Context*

PRE TEST KEY Salmon Fishing: Investigations into A 6 th grade module in the Math in a Cultural Context* UNIVERSITY OF ALASKA FAIRBANKS Student Name: PRE TEST KEY Grade: Teacher: School: Location of School:

### 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

### 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

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

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

### Probability and Statistics 15% of EOC

MGSE9-12.S.CP.1 1. Which of the following is true for A U B A: 2, 4, 6, 8 B: 5, 6, 7, 8, 9, 10 A. 6, 8 B. 2, 4, 6, 8 C. 2, 4, 5, 6, 6, 7, 8, 8, 9, 10 D. 2, 4, 5, 6, 7, 8, 9, 10 2. This Venn diagram shows

### Independent Events B R Y

. Independent Events Lesson Objectives Understand independent events. Use the multiplication rule and the addition rule of probability to solve problems with independent events. Vocabulary independent

### Math 3201 Unit 3: Probability Name:

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

### 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

### 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

### Section 6.5 Conditional Probability

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

### Probability Warm-Up 2

Probability Warm-Up 2 Directions Solve to the best of your ability. (1) Write out the sample space (all possible outcomes) for the following situation: A dice is rolled and then a color is chosen, blue

### Probability and Randomness. Day 1

Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of

### 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,

### Name: 1. Match the word with the definition (1 point each - no partial credit!)

Chapter 12 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. SHOW ALL YOUR WORK!!! Remember

### Math 1 Unit 4 Mid-Unit Review Chances of Winning

Math 1 Unit 4 Mid-Unit Review Chances of Winning Name My child studied for the Unit 4 Mid-Unit Test. I am aware that tests are worth 40% of my child s grade. Parent Signature MM1D1 a. Apply the addition

### 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

### 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

### 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.

### Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ.

Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ. 1 Chance of winning?! What s the chances of winning New York Megamillion Jackpot!! just pick 5 numbers from 1 to 56, plus a mega ball number

### 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

### CONDITIONAL PROBABILITY Assignment

State which the following events are independent and which are dependent.. Drawing a card from a standard deck of playing card and flipping a penny 2. Drawing two disks from an jar without replacement

### 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

### HARDER PROBABILITY. Two events are said to be mutually exclusive if the occurrence of one excludes the occurrence of the other.

HARDER PROBABILITY MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION LAW OF PROBABILITY Two events are said to be mutually exclusive if the occurrence of one excludes the occurrence of the other. Example Throwing

### What s the Probability I Can Draw That? Janet Tomlinson & Kelly Edenfield

What s the Probability I Can Draw That? Janet Tomlinson & Kelly Edenfield Engage Your Brain On your seat you should have found a list of 5 events and a number line on which to rate the probability of those

### 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

### Mutually Exclusive Events Algebra 1

Name: Mutually Exclusive Events Algebra 1 Date: Mutually exclusive events are two events which have no outcomes in common. The probability that these two events would occur at the same time is zero. Exercise

### LISTING THE WAYS. getting a total of 7 spots? possible ways for 2 dice to fall: then you win. But if you roll. 1 q 1 w 1 e 1 r 1 t 1 y

LISTING THE WAYS A pair of dice are to be thrown getting a total of 7 spots? There are What is the chance of possible ways for 2 dice to fall: 1 q 1 w 1 e 1 r 1 t 1 y 2 q 2 w 2 e 2 r 2 t 2 y 3 q 3 w 3

### 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 MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37

Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete

### Discrete Structures for Computer Science

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

### 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.

### Functional Skills Mathematics

Functional Skills Mathematics Level Learning Resource Probability D/L. Contents Independent Events D/L. Page - Combined Events D/L. Page - 9 West Nottinghamshire College D/L. Information Independent Events

### 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

### STATISTICAL COUNTING TECHNIQUES

STATISTICAL COUNTING TECHNIQUES I. Counting Principle The counting principle states that if there are n 1 ways of performing the first experiment, n 2 ways of performing the second experiment, n 3 ways

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

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

### 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

### 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 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

### 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

### PROBABILITY.0 Concept Map Contents Page. Probability Of An Event. Probability Of Two Events. 4. Probability of Mutually Exclusive Events.4 Probability

PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM MODULE PROBABILITY PROBABILITY.0 Concept Map Contents Page. Probability Of An Event. Probability Of Two Events. 4. Probability of Mutually