MAT Mathematics in Today's World


 Ronald Day
 2 years ago
 Views:
Transcription
1 MAT 1000 Mathematics in Today's World
2 We talked about Cryptography Last Time
3 We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities is to make a list of everything that can happen this is called a sample space. A probability model is the sample space with probabilities for each outcome.
4 Random Phenomena A phenomenon is random if each individual occurrence is uncertain, but there is a pattern over many occurences. Examples of random phenomena: Tossing a coin, rolling a die, drawing a card. The gender of a baby.
5 Random Phenomena To describe random phenomena, start by giving a list of everything that could happen. Each distinct thing that could happen is called an outcome. If we toss a coin, there are two outcomes: heads or tails. We can t predict which will happen, but in the long run 50% of the tosses are heads and 50% are tails.
6 Random Phenomena If we roll a (sixsided) die, there are six possible outcomes: We can t predict in advance which outcome will occur. In the long run each outcome will occur 1/6 th of the time.
7 Random Phenomena We don t know in advance if a baby will be a boy or a girl. In the long run it turns out that about 49% of newborns are girls, and 51% are boys. Note that even though there are two outcomes, they don t have to occur in equal proportions.
8 Random Phenomena To describe random phenomena, start by giving a list of everything that could happen. Each distinct thing that could happen is called an outcome. Each outcome of a random phenomenon has a probability. The probability of an outcome is the proportion of times it occurs over many trials. Probabilities are proportions, so we can express them as percents (50%), decimals (0.5), or fractions (1/2), whichever is most convenient. I will use all of these.
9 Random Phenomena We can restate our earlier observations in terms of probabilities: When we flip a coin the probability of getting tails is 50%. When we roll a die, the probability of getting a 4 is 1/6 The probability that a newborn baby is a girl is about 49%
10 Sample Space When there are not too many outcomes, one good way to analyze a random phenomenon is to make a list of every outcome. The set of all of the outcomes of a random phenomenon is called the sample space. For tossing a coin the sample space is {heads, tails} For rolling a die the sample space is {1, 2, 3, 4, 5, 6}
11 Random Phenomena We use the term event to describe a subset of the sample space. For example, when we roll a die, we can look for the event: rolling a odd number. There are three outcomes in this event: rolling a 1, a 3, or a 5. As a set, this event is {1, 3, 5} The probability of this event is 1 2
12 Random Phenomena Another event is rolling a 6 There is only one outcome in this event. As a set, this it is {6} The probability of this event is 1 6
13 Probability models The sample space tells us all of the possible outcomes. If we include the probability of each outcome we get what is called a probability model. Example Here is the probability model for tossing a coin: Heads Tails
14 Probability models Here is a probability model for the gender of a newborn: Boy Girl Here is a probability model for tossing a coin: /6 1/6 1/6 1/6 1/6 1/6 The top row is a list of the outcomes. The bottom row is the probability of each.
15 Probability Rules There are four rules which govern probabilities. Any probability is a number between 0 and 1. A probability of an event is the proportion of times that it can be expected to occur. The larger the probability, the more likely it is.
16 Probability Rules If the probability of an event is 1, it will always happen for sure. If the probability of an event is 0, it never happens. Note that neither of these are really random!
17 Probability Rules The probability that an event does not occur is 1 minus the probability that the event does occur. What is the probability that we roll a die and do not get a 6? The probability of getting a 6 is 1 6. By the rule, the probability of not getting a 6 should be = 5 6
18 Probability Rules If two events have no outcomes in common, they are said to be disjoint. The probability that one or the other of two disjoint events occurs is the sum of their individual probabilities. The two events rolling an odd number and rolling a 6 are the sets {1, 3, 5} {6} These sets are disjoint (nothing in common).
19 Probability Rules By the rule, the probability that we roll either an odd number or a 6 is the sum of their individual probabilities: To add these fractions we can use a common denominator of 6 3 This can be reduced = = 2 3
20 Probability Rules All possible outcomes together must have probability 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly one.
21 Probability models We can use this last rule to fill in missing information in a probability model. If you draw an M&M candy at random from the bag, it will have one of six colors. Here is a probability model: What is the probability of randomly drawing a blue M&M? The probabilities of all the outcomes must add up to 1. So Brown Red Yellow Green Orange Blue ? Blue = 1 The probability the M&M is blue is 0.10
22 Independent Events If the outcome of one event has no impact on the outcome of the other, we say the two events are independent. Each roll of a die is independent from any other roll. Why? Because the outcome of one roll does not affect the next roll. The weather one day is not independent from the weather the next day. Why? Because weather moves in systems: if it rains one day, the probability of it raining the next day is increased.
23 Independent Events The most important fact about independent events is that the probability that both happen is the product of the individual probabilities. The probability that I get heads on a coin toss is 1/2. The probability that I get a 1 when I roll a die is 1/6. These are independent events. So if I toss a coin then roll a die, the probability that I get heads on the coin and a 1 on the die is the product: = 1 12
24 Probability models We are now going to put all of this information together to start constructing more complicated probability models. Remember a probability model is a list of outcomes and their probabilities.
25 Probability models If we toss two coins, what are the possible outcomes? We can get heads on both, tails on both, or heads on one and tails on the other. But be careful, there are actually four outcomes. Why? This is easiest to see if you imagine the coins being different, say one is a quarter and one is a dime: Quarter H H T T Dime H T H T
26 Probability models Now we have a sample space with four outcomes. Let s make a probability model. What is the probability of getting heads on each coin? The coin tosses are independent, so probability of heads on the quarter and heads on the dime = (probability of heads on the quarter) X (probability of heads on the dime) The probability of getting heads on both is = 0.25 In a similar way the probability of each of the other three outcomes is also 0.25
27 Probability models So the probability model for tossing two coins is: H,H T,H H,T T,T What is the probability of tossing two coins and getting one head and one tail? This is an event, containing two outcomes: T,H and H,T So the probability of one head and one tail is = 0.50
28 Probability models Here s a related example: let s find a probability model for rolling two dice. First we should make a sample space: Again we distinguish getting a 3 on the first die and a 4 on the second from getting a 4 on the first and a 3 on the second.
29 Probability models It is useful to write outcomes here as pairs of numbers. So (3,4) indicates a 3 on the first die and a 4 on the second. As I said on the last slide: the outcome (3,4) is not the same as the outcome (4,3). Now we know the sample space. What are the probabilities of each outcome?
30 Probability models It turns out that each outcome has the same probability: How many outcomes are there? 36. They all have the same probability, so each one has probability 1/36.
31 Probability models Now, suppose we roll 2 dice and then add the numbers we get. So if we get (2,5), the outcome is the sum of these What is the sample space? = 7 The possible outcomes (sums of two numbers between 1 and 6) are {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
32 Probability models What s the probability of each of these outcomes? Look back at the sample space for rolling two dice: We get a sum of 2 only from the outcome (1,1). The probability is 1 36 There are two ways to get a sum of 3: (1,2) and (2,1). The probability is = 2 36 = 1 18
33 Probability models How many ways are there to get a sum of 4? There are three ways: (1,3), (2,2), or (3,1). The probability of getting a sum of 4 is: = 3 36 = 1 12 Continuing in this way, we get the following probability model: Outcome Probability = 1 18 = 1 12 = 1 9 = 1 6 = 1 9 = 1 12 = 1 18
34 Probability models It can be useful to represent probability models using probability histograms. The horizontal scale gives the outcomes, and the probability of each outcome determines the height of the bar:
Events. Independence. Coin Tossing. Random Phenomena
Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,
More informationAP 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
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationQuestion: What is the probability that a fivecard 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
More informationAn event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event
An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the
More informationProbability Review. ICPSR Applied Bayesian Modeling
Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How
More informationMATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics
MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you should
More informationMathematical 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
More informationMost of us would probably believe they are the same, it would not make a difference. But, in fact, they are different. Let s see how.
PROBABILITY If someone told you the odds of an event A occurring are 3 to 5 and the probability of another event B occurring was 3/5, which do you think is a better bet? Most of us would probably believe
More informationStatistical 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
More information+ 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
More informationChapter 4: Probabilities and Proportions
Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,
More informationContemporary 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
More informationProbabilities of Compound Events
0 LESSON Probabilities of Compound Events UNDERSTAND Sometimes, you may want to find the probability that two or more events will occur at the same time. This is called finding the probability of a compound
More informationBasic Probability Theory I
A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationDistributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment
C Probability and Probability Distributions APPENDIX C.1 Probability A1 C.1 Probability Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment When assigning
More informationSection 65 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)
More informationName: Date: Use the following to answer questions 24:
Name: Date: 1. A phenomenon is observed many, many times under identical conditions. The proportion of times a particular event A occurs is recorded. What does this proportion represent? A) The probability
More informationnumber 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.
More informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair sixsided 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
More informationI. 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
More information1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.
1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first
More informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More informationPROBABILITY. 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
More informationProbability (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
More informationSample Space, Events, and PROBABILITY
Sample Space, Events, and PROBABILITY In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More information33 Probability: Some Basic Terms
33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been
More informationChapter 5: Probability: What are the Chances? Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation
Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 5 Probability: What Are
More informationHomework 5 Solutions
Math 130 Assignment Chapter 18: 6, 10, 38 Chapter 19: 4, 6, 8, 10, 14, 16, 40 Chapter 20: 2, 4, 9 Chapter 18 Homework 5 Solutions 18.6] M&M s. The candy company claims that 10% of the M&M s it produces
More informationChapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.
MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.
More informationChapter 4 Probability
The Big Picture of Statistics Chapter 4 Probability Section 42: Fundamentals Section 43: Addition Rule Sections 44, 45: Multiplication Rule Section 47: Counting (next time) 2 What is probability?
More informationThe Central Limit Theorem Part 1
The Central Limit Theorem Part. Introduction: Let s pose the following question. Imagine you were to flip 400 coins. To each coin flip assign if the outcome is heads and 0 if the outcome is tails. Question:
More informationUnit 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,
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More information2. 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
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationExample: Use the Counting Principle to find the number of possible outcomes of these two experiments done in this specific order:
Section 4.3: Tree Diagrams and the Counting Principle It is often necessary to know the total number of outcomes in a probability experiment. The Counting Principle is a formula that allows us to determine
More informationH + T = 1. p(h + T) = p(h) x p(t)
Probability and Statistics Random Chance A tossed penny can land either heads up or tails up. These are mutually exclusive events, i.e. if the coin lands heads up, it cannot also land tails up on the same
More informationPROBABILITY NOTIONS. Summary. 1. Random experiment
PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non
More informationChapter 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
More informationV. 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
More informationWorked 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
More informationSession 8 Probability
Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome
More informationBasic 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
More informationProbability experiment. Toss two coins Toss three coins Roll two dice
Probability experiment Toss two coins Toss three coins Roll two dice Activity 1: An Experiment with 2 coins WE TOSS TWO COINS When we toss two coins at the same time, the possible outcomes are: (two Heads)
More informationLesson 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.
More informationChapter 13 & 14  Probability PART
Chapter 13 & 14  Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph
More informationMathematics Content: Pie Charts; Area as Probability; Probabilities as Percents, Decimals & Fractions
Title: Using the Area on a Pie Chart to Calculate Probabilities Mathematics Content: Pie Charts; Area as Probability; Probabilities as Percents, Decimals & Fractions Objectives: To calculate probability
More informationProbability, statistics and football Franka Miriam Bru ckler Paris, 2015.
Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups
More informationIt is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important
PROBABILLITY 271 PROBABILITY CHAPTER 15 It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important subject of human knowledge.
More informationProbability: 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
More informationPROBABILITY 14.3. section. The Probability of an Event
4.3 Probability (43) 727 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
More informationFormula for Theoretical Probability
Notes Name: Date: Period: Probability I. Probability A. Vocabulary is the chance/ likelihood of some event occurring. Ex) The probability of rolling a for a sixfaced die is 6. It is read as in 6 or out
More information94 Counting Solutions for Chapter 3. Section 3.2
94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length4 lists are there? Answer: 6 6 6 6 = 1296. (b
More informationProbability Lesson #2
Probability Lesson #2 Sample Space A sample space is the set of all possible outcomes of an experiment. There are a variety of ways of representing or illustrating sample spaces. Listing Outcomes List
More informationConsider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.
Probability and the ChiSquare Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system
More informationUnit 18: Introduction to Probability
Unit 18: Introduction to Probability Summary of Video There are lots of times in everyday life when we want to predict something in the future. Rather than just guessing, probability is the mathematical
More informationLesson 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
More informationDetermining Probabilities Using Tree Diagrams and Tables
Determining Probabilities Using ree Diagrams and ables Focus on After this lesson, you will be able to φ determine the φ φ sample space of a probability experiment with two independent events represent
More informationComputer Skills Microsoft Excel Creating Pie & Column Charts
Computer Skills Microsoft Excel Creating Pie & Column Charts In this exercise, we will learn how to display data using a pie chart and a column chart, colorcode the charts, and label the charts. Part
More informationThe 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
More informationThis is Basic Concepts of Probability, chapter 3 from the book Beginning Statistics (index.html) (v. 1.0).
This is Basic Concepts of Probability, chapter 3 from the book Beginning Statistics (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationFraction Decimal Percent a) 4 5
Fractions, Decimals and Percents To convert a fraction to a percent, convert the fraction to a decimal number by dividing the numerator by the denominator. Then, multiply the decimal by 100 and add a percent
More informationChapter. Probability Pearson Education, Inc. All rights reserved. 1 of 20
Chapter 3 Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 20 Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition
More informationChapter 14 From Randomness to Probability
Chapter 14 From Randomness to Probability 199 Chapter 14 From Randomness to Probability 1. Sample spaces. a) S = { HH, HT, TH, TT} All of the outcomes are equally likely to occur. b) S = { 0, 1, 2, 3}
More informationBasics of Probability
Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.
More informationSTT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012)
STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012) TA: Zhen (Alan) Zhang zhangz19@stt.msu.edu Office hour: (C500 WH) 1:45 2:45PM Tuesday (office tel.: 4323342) Helproom: (A102 WH) 11:20AM12:30PM,
More informationYou flip a fair coin four times, what is the probability that you obtain three heads.
Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.
More information7.1 Sample space, events, probability
7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationGrade 7 Mathematics. Unit 7. Data Analysis. Estimated Time: 18 Hours
Grade 7 Mathematics Data Analysis Estimated Time: 18 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization
More informationA (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.
Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome
More informationPROBABILITY. 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
More informationConducting Probability Experiments
CHAPTE Conducting Probability Experiments oal Compare probabilities in two experiments. ame. Place a shuffled deck of cards face down.. Turn over the top card.. If the card is an ace, you get points. A
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationSection 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
More informationChapter 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
More informationMath 3C Homework 3 Solutions
Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard
More informationActivities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median
Activities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median 58 What is a Ratio? A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a
More informationThe basics of probability theory. Distribution of variables, some important distributions
The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a
More information7.S.8 Interpret data to provide the basis for predictions and to establish
7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an
More informationSTRAND D: PROBABILITY. UNIT D2 Probability of Two or More Events: Text. Contents. Section. D2.1 Outcome of Two Events. D2.2 Probability of Two Events
PRIMARY Mathematics SKE: STRAND D STRAND D: PROAILITY D2 Probability of Two or More Events * Text Contents * * * Section D2. Outcome of Two Events D2.2 Probability of Two Events D2. Use of Tree Diagrams
More informationBasic 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.
More informationProbability. 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
More information1 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
More informationChapter 5  Probability
Chapter 5  Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set
More informationDecision 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
More informationToss a coin twice. Let Y denote the number of heads.
! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.
More informationChapter 10: Introducing Probability
Chapter 10: Introducing Probability Randomness and Probability So far, in the first half of the course, we have learned how to examine and obtain data. Now we turn to another very important aspect of Statistics
More informationProbability and Expected Value
Probability and Expected Value Lesson Plan Cube Fellow: Audrey Brock Teacher Mentor: Karyl White Goal: The goal of this lesson is to introduce students to the methods and ideas behind probability theory.
More informationSection 6.2 ~ Basics of Probability. Introduction to Probability and Statistics SPRING 2016
Section 6.2 ~ Basics of Probability Introduction to Probability and Statistics SPRING 2016 Objective After this section you will know how to find probabilities using theoretical and relative frequency
More informationPROBABILITY SECOND EDITION
PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All
More informationDecimal and Fraction Review Sheet
Decimal and Fraction Review Sheet Decimals Addition To add 2 decimals, such as 3.25946 and 3.514253 we write them one over the other with the decimal point lined up like this 3.25946 +3.514253 If one
More informationChapter 4: Probability
Chapter 4: Probability Section 4.1: Empirical Probability One story about how probability theory was developed is that a gambler wanted to know when to bet more and when to bet less. He talked to a couple
More informationMath Games For Skills and Concepts
Math Games p.1 Math Games For Skills and Concepts Original material 20012006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,
More informationSTAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia
STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that
More informationAbout chance. Likelihood
Chance deals with the concepts of randomness and the use of probability as a measure of how likely it is that particular events will occur. A National Statement on Mathematics for Australian Schools, 1991
More information4.5 Finding Probability Using Tree Diagrams and Outcome Tables
4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or
More informationIn 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
More information