# 1MA01: Probability. Sinéad Ryan. November 12, 2013 TCD

Size: px
Start display at page:

Transcription

1 1MA01: Probability Sinéad Ryan TCD November 12, 2013

2 Definitions and Notation EVENT: a set possible outcomes of an experiment. Eg flipping a coin is the experiment, landing on heads is the event If an experiment is repeated many times then number of times event A happens number of times experiment repeated is called the Probability of event A, written P(A)

3 An example: Roll a die 1000 times and record the frequency with which each number appears giving something like VALUE FREQ P(1) = = P(2) = = =. Note that 6 P(i) = 1. i=1 Note that if P is a probability, 0 P 1

4 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red?

5 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20

6 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white?

7 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white? P(white) = 0 20 = 0

8 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white? P(white) = 0 20 = 0 the ball is either black, yellow, green or red?

9 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white? P(white) = 0 20 = 0 the ball is either black, yellow, green or red? P(b, y, g, r) = = 1

10 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace?

11 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13

12 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king?

13 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13

14 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king?

15 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king? P(not a king) = = 12 13

16 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king? P(not a king) = = drawing an ace or a king?

17 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack - drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king? P(not a king) = = drawing an ace or a king? P(ace or king) = P(ace) + P(king) = 2 13

18 The assumption of disjoint events is crucial for addition of probabilities. So, eg. drawing an ace and drawing a heart are not disjoint so addition rules don t hold ie. P(ace or heart) = P(ace) + P(heart) Note above that P(king) = 1/ 13 and P(not king) = 12/ 13. These disjoint events are complements and since exactly one of these must happen then their probabilities sum to 1 P(king) + P(not king) = 1 P(not king) = 1 P(king) In general then, if A is an event then P(not A) = 1 P(A).

19 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart?

20 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4

21 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?

22 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?p(b A) = = 4 17

23 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?p(b A) = = 4 17 What is the probability both cards are hearts?

24 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?p(b A) = = 4 17 What is the probability both cards are hearts? P(A and B) = P(A)P(B A) = 1 17

25 In the last example you could also use the P.E.L.O. and list all 2652 ways the 1st and 2nd cards could be dealt. Clearly using the multiplication rule is easier. Independence Events A and B are independent if P(B A) = P(B) and in this case P(A and B) = P(A)P(B). Note that independence is not the same thing as disjoint. A simple example of independence: roll a die twice. What is the probability of rolling 2 sixes? P(six and six) = P(A)P(B A) = P(A)P(B) = = 1 36 since the result of each roll is independent of the other. 1

### Objectives. Determine whether events are independent or dependent. Find the probability of independent and dependent events.

Objectives Determine whether events are independent or dependent. Find the probability of independent and dependent events. independent events dependent events conditional probability Vocabulary Events

### Probability Review 41

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

### Objective 1: Simple Probability

Objective : Simple Probability To find the probability of event E, P(E) number of ways event E can occur total number of outcomes in sample space Example : In a pet store, there are 5 puppies, 22 kittens,

### Probability 1. Joseph Spring School of Computer Science. SSP and Probability

Probability 1 Joseph Spring School of Computer Science SSP and Probability Areas for Discussion Experimental v Theoretical Probability Looking Back v Looking Forward Theoretical Probability Sample Space,

### 4.3 Some Rules of Probability

4.3 Some Rules of Probability Tom Lewis Fall Term 2009 Tom Lewis () 4.3 Some Rules of Probability Fall Term 2009 1 / 6 Outline 1 The addition rule 2 The complement rule 3 The inclusion/exclusion principle

### 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

### 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

### 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

### (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

### Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules

+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that

### Lesson 3 Dependent and Independent Events

Lesson 3 Dependent and Independent Events When working with 2 separate events, we must first consider if the first event affects the second event. Situation 1 Situation 2 Drawing two cards from a deck

### 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

### 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. 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

### 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

### Textbook: pp Chapter 2: Probability Concepts and Applications

1 Textbook: pp. 39-80 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.

### APPENDIX 2.3: RULES OF PROBABILITY

The frequentist notion of probability is quite simple and intuitive. Here, we ll describe some rules that govern how probabilities are combined. Not all of these rules will be relevant to the rest of this

### 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

### 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

### 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.

### 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

### 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

### Probability I Sample spaces, outcomes, and events.

Probability I Sample spaces, outcomes, and events. When we perform an experiment, the result is called the outcome. The set of possible outcomes is the sample space and any subset of the sample space is

### Elementary Statistics. Basic Probability & Odds

Basic Probability & Odds What is a Probability? Probability is a branch of mathematics that deals with calculating the likelihood of a given event to happen or not, which is expressed as a number between

### 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

### Probability: Terminology and Examples Spring January 1, / 22

Probability: Terminology and Examples 18.05 Spring 2014 January 1, 2017 1 / 22 Board Question Deck of 52 cards 13 ranks: 2, 3,..., 9, 10, J, Q, K, A 4 suits:,,,, Poker hands Consists of 5 cards A one-pair

### 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

### When a number cube is rolled once, the possible numbers that could show face up are

C3 Chapter 12 Understanding Probability Essential question: How can you describe the likelihood of an event? Example 1 Likelihood of an Event When a number cube is rolled once, the possible numbers that

### 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,

### Math 14 Lecture Notes Ch. 3.3

3.3 Two Basic Rules of Probability If we want to know the probability of drawing a 2 on the first card and a 3 on the 2 nd card from a standard 52-card deck, the diagram would be very large and tedious

### Classical vs. Empirical Probability Activity

Name: Date: Hour : Classical vs. Empirical Probability Activity (100 Formative Points) For this activity, you will be taking part in 5 different probability experiments: Rolling dice, drawing cards, drawing

### 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

### 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

### 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

### Chapter 16. Probability. For important terms and definitions refer NCERT text book. (6) NCERT text book page 386 question no.

Chapter 16 Probability For important terms and definitions refer NCERT text book. Type- I Concept : sample space (1)NCERT text book page 386 question no. 1 (*) (2) NCERT text book page 386 question no.

### November 11, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.

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

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

### 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

### 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

### 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,

### Probability Test Review Math 2. a. What is? b. What is? c. ( ) d. ( )

Probability Test Review Math 2 Name 1. Use the following venn diagram to answer the question: Event A: Odd Numbers Event B: Numbers greater than 10 a. What is? b. What is? c. ( ) d. ( ) 2. In Jason's homeroom

### 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.

### 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:

### 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

### Important Distributions 7/17/2006

Important Distributions 7/17/2006 Discrete Uniform Distribution All outcomes of an experiment are equally likely. If X is a random variable which represents the outcome of an experiment of this type, then

### CS 361: Probability & Statistics

February 7, 2018 CS 361: Probability & Statistics Independence & conditional probability Recall the definition for independence So we can suppose events are independent and compute probabilities Or we

### 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

### 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.

### 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

### STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving.

Worksheet 4 th Topic : PROBABILITY TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving. BASIC COMPETENCY:

### Module 4 Project Maths Development Team Draft (Version 2)

5 Week Modular Course in Statistics & Probability Strand 1 Module 4 Set Theory and Probability It is often said that the three basic rules of probability are: 1. Draw a picture 2. Draw a picture 3. Draw

### 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

### Counting integral solutions

Thought exercise 2.2 20 Counting integral solutions Question: How many non-negative integer solutions are there of x 1 +x 2 +x 3 +x 4 = 10? Thought exercise 2.2 20 Counting integral solutions Question:

### 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)

### heads 1/2 1/6 roll a die sum on 2 dice 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 1, 2, 3, 4, 5, 6 heads tails 3/36 = 1/12 toss a coin trial: an occurrence

trial: an occurrence roll a die toss a coin sum on 2 dice sample space: all the things that could happen in each trial 1, 2, 3, 4, 5, 6 heads tails 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 example of an outcome:

### 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,

### 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

### Math 102 Practice for Test 3

Math 102 Practice for Test 3 Name Show your work and write all fractions and ratios in simplest form for full credit. 1. If you draw a single card from a standard 52-card deck what is P(King face card)?

### Lecture 6 Probability

Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times

### Most of the time we deal with theoretical probability. Experimental probability uses actual data that has been collected.

AFM Unit 7 Day 3 Notes Theoretical vs. Experimental Probability Name Date Definitions: Experiment: process that gives a definite result Outcomes: results Sample space: set of all possible outcomes Event:

### When combined events A and B are independent:

A Resource for ree-standing Mathematics Qualifications A or B Mutually exclusive means that A and B cannot both happen at the same time. Venn Diagram showing mutually exclusive events: Aces The events

### 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.

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,

### 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

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

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

### Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

### 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

### Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

### 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

### 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

### Simple Probability. Arthur White. 28th September 2016

Simple Probability Arthur White 28th September 2016 Probabilities are a mathematical way to describe an uncertain outcome. For eample, suppose a physicist disintegrates 10,000 atoms of an element A, and

### Chapter 4: Introduction to Probability

MTH 243 Chapter 4: Introduction to Probability Suppose that we found that one of our pieces of data was unusual. For example suppose our pack of M&M s only had 30 and that was 3.1 standard deviations below

### Sample Spaces, Events, Probability

Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014 Sets A set is a collection of unique objects. Sets A set is a collection of unique objects.

### CHAPTERS 14 & 15 PROBABILITY STAT 203

CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical

### Before giving a formal definition of probability, we explain some terms related to probability.

probability 22 INTRODUCTION In our day-to-day life, we come across statements such as: (i) It may rain today. (ii) Probably Rajesh will top his class. (iii) I doubt she will pass the test. (iv) It is unlikely

### Revision 6: Similar Triangles and Probability

Revision 6: Similar Triangles and Probability Name: lass: ate: Mark / 52 % 1) Find the missing length, x, in triangle below 5 cm 6 cm 15 cm 21 cm F 2) Find the missing length, x, in triangle F below 5

### 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

### TEST A CHAPTER 11, PROBABILITY

TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability

### Chapter-wise questions. Probability. 1. Two coins are tossed simultaneously. Find the probability of getting exactly one tail.

Probability 1. Two coins are tossed simultaneously. Find the probability of getting exactly one tail. 2. 26 cards marked with English letters A to Z (one letter on each card) are shuffled well. If one

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

Chapter 6: Probability and Simulation The study of randomness Introduction Probability is the study of chance. 6.1 focuses on simulation since actual observations are often not feasible. When we produce

### Skills we've learned. Skills we need. 7 3 Independent and Dependent Events. March 17, Alg2 Notes 7.3.notebook

7 3 Independent and Dependent Events Skills we've learned 1. In a box of 25 switches, 3 are defective. What is the probability of randomly selecting a switch that is not defective? 2. There are 12 E s

### Introduction to Probability and Statistics I Lecture 7 and 8

Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes

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

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

### the total number of possible outcomes = 1 2 Example 2

6.2 Sets and Probability - A useful application of set theory is in an area of mathematics known as probability. Example 1 To determine which football team will kick off to begin the game, a coin is tossed

### 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

### 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.

### Probability and the Monty Hall Problem Rong Huang January 10, 2016

Probability and the Monty Hall Problem Rong Huang January 10, 2016 Warm-up: There is a sequence of number: 1, 2, 4, 8, 16, 32, 64, How does this sequence work? How do you get the next number from the previous

### Lesson 16.1 Assignment

Lesson 16.1 Assignment Name Date Rolling, Rolling, Rolling... Defining and Representing Probability 1. Rasheed is getting dressed in the dark. He reaches into his sock drawer to get a pair of socks. He

### 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

### Diamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES

CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times

### A referee flipped a fair coin to decide which football team would start the game with

Probability Lesson.1 A referee flipped a fair coin to decide which football team would start the game with the ball. The coin was just as likely to land heads as tails. Which way do you think the coin

### 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:

### Compound Probability. A to determine the likelihood of two events occurring at the. ***Events can be classified as independent or dependent events.

Probability 68B A to determine the likelihood of two events occurring at the. ***Events can be classified as independent or dependent events. Independent Events are events in which the result of event

### 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,

### 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.

### 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

### Introduction to probability

Introduction to probability Suppose an experiment has a finite set X = {x 1,x 2,...,x n } of n possible outcomes. Each time the experiment is performed exactly one on the n outcomes happens. Assign each