Stat 20: Intro to Probability and Statistics

Size: px
Start display at page:

Transcription

1 Stat 20: Intro to Probability and Statistics Lecture 17: Using the Normal Curve with Box Models Tessa L. Childers-Day UC Berkeley 23 July 2014

2 By the end of this lecture... You will be able to: Draw and interpret a probability histogram for the sum draws from a box Predict what a probability histogram for the sum of draws from a box will look like Use the Central Limit Theorem to calculate probabilities without having to draw a histogram 2 / 19

3 Recap: Box Models & The Normal Curve Box models are useful in analyzing games of chance Draw a box model to describe a process (sums or classify/count) Calculate EV and SE Use normal curve to calculate probability that actually observe some event But why use normal curve? 3 / 19

4 If you have enough draws/trials... The chances become normal. We ll see this via experimentation. Say we are flipping a fair coin. What does our box look like? What if we want to count the number of heads? What is P(Heads)? P(Tails)? What about if there are 2 tosses? What about more tosses? 4 / 19

5 Summarize a probability table A histogram will summarize the probabilities well. What should go on the x-axis? What should our classes be? What about the y-axis? 5 / 19

6 Summarize a probability table (cont.) What did our experiment show us? As the number of draws, the becomes more like a normal curve Probability histograms represent Height represents percent per Make probability histogram more normal by 6 / 19

7 What if the box changes? With the coin flips, the probability histogram for the number of heads began to look normal fairly quickly. What about other scenarios? Add another face, \$1 for head, \$0 for tail, \$15 for new face Sum of dice Number of 6 s Net gain on bet that pays 2 to 1. Win if a single draw from a deck is black. Net gain on bet that pays 25 to 1. Win if a single draw from a deck is a Queen. Others? 7 / 19

8 What have we learned? Does this work for any function of the draws? Does this work for any box? Does this work for any number of draws? How many draws do we need? 8 / 19

9 Probability Histogram Normal Curve With enough draws, the probability histogram for the sum of draws looks normal. Why is this useful? To use the normal curve we need: z = value center spread Value Center Spread 9 / 19

10 The Value(s) Draw a picture, shade an area What do we want to draw? What do we want to shade? Why do we shade this area? When do we use this technique? 10 / 19

11 The Center Draw a picture, shade an area. Where is the center of our picture? Is our picture symmetric? How big is a typical draw? How big is a typical sum? 11 / 19

12 The Spread Draw a picture, shade an area. How spread out is our picture? How far from the typical draw is an individual draw going to be (on average)? How far from the typical sum is an individual sum going to be (on average)? 12 / 19

13 Put it all together Want the chance a sum is between val 1 and val 2 1 Draw a picture, shade area between val and val Convert to z-scores: z 1 = (val 1.5) EV for sum SE for sum and z 2 = (val 2 +.5) EV for sum SE for sum 3 Use normal table to find area 13 / 19

14 The CLT Just a fancy way of saying what we already know. When drawing a random, with replacement, from a box, if there are enough draws, the probability histogram looks normal. The associated normal curve: is centered at the EV for the sum has spread equal to the SE for the sum 14 / 19

15 Example: Playing the Slots A slot machine contains 3 drums that turn rapidly, and then stop on a random character. Each drum has 3 characters: a bar, a cherry, and a diamond. The first drum contains 5 characters (a bar, 2 cherries, and a diamond). The second drum contains 6 characters (2 bars, 3 cherries, and a diamond). The third drum contains 7 characters (3 bars, 4 cherries, and a diamond). What is the chance that in one spin you see all 3 diamonds? What is the chance that in one spin you see all 3 characters the same? 15 / 19

16 Example: Playing the Slots (cont.) A slot machine contains 3 drums that turn rapidly, and then stop on a random character. First drum: a bar, 2 cherries, and a diamond. Second drum: 2 bars, 3 cherries, and a diamond. Third drum: 3 bars, 4 cherries, and a diamond. Suppose it costs \$5 to play the slot machine. The machine pays \$100 if you see all 3 diamonds. It pays \$10 if you see either all 3 bars or all 3 cherries. Make a box model for your net gain when playing this machine. Make a box model for counting the number of times you win when playing this machine. 16 / 19

17 Example: Playing the Slots (cont.) A slot machine contains 3 drums that turn rapidly, and then stop on a random character. First drum: a bar, 2 cherries, and a diamond. Second drum: 2 bars, 3 cherries, and a diamond. Third drum: 3 bars, 4 cherries, and a diamond. You decide to play this game 10 times. Relate your net gain for 10 plays to a box model. About how much do you expect to win? About how much is this likely to be off by? Calculate the chance that you lose between \$25 and \$50 playing 10 times. Do you think your calculation is accurate? 17 / 19

18 Example: Playing the Slots (cont.) A slot machine contains 3 drums that turn rapidly, and then stop on a random character. First drum: a bar, 2 cherries, and a diamond. Second drum: 2 bars, 3 cherries, and a diamond. Third drum: 3 bars, 4 cherries, and a diamond. You and a group of 9 friends decide to play on similar machines. You each play 100 times. Relate your net gain (as a group) to a box model. About how much do you expect to win (as a group)? About how much is this likely to be off by? Calculate the chance that you lose between \$2500 and \$3000 playing as a group. Do you think your calculation is accurate? 18 / 19

19 Important Takeaways A probability histogram displays probabilities, not data With enough draws, a probability histogram for the sum of draws with replacement looks like a normal curve How big is enough? Influenced by the make-up of the box The normal curve is centered at the EV for the sum, with spread equal to the SE for the sum We can use the continuity correction, combined with the normal curve, to find probabilities under the normal curve (approximate probabilities from the probability histogram) Next time: What if we re not gambling, but sampling? 19 / 19

Grade 8 Math Assignment: Probability Part 1: Rock, Paper, Scissors - The Study of Chance Purpose An introduction of the basic information on probability and statistics Materials: Two sets of hands Paper

There is no class tomorrow! Have a good weekend! Scores will be posted in Compass early Friday morning J

STATISTICS 100 EXAM 3 Fall 2016 PRINT NAME (Last name) (First name) *NETID CIRCLE SECTION: L1 12:30pm L2 3:30pm Online MWF 12pm Write answers in appropriate blanks. When no blanks are provided CIRCLE your

a) Getting 10 +/- 2 head in 20 tosses is the same probability as getting +/- heads in 320 tosses

Question 1 pertains to tossing a fair coin (8 pts.) Fill in the blanks with the correct numbers to make the 2 scenarios equally likely: a) Getting 10 +/- 2 head in 20 tosses is the same probability as

Math 147 Lecture Notes: Lecture 21

Math 147 Lecture Notes: Lecture 21 Walter Carlip March, 2018 The Probability of an Event is greater or less, according to the number of Chances by which it may happen, compared with the whole number of

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Lecture 12: More Probability Tessa L. Childers-Day UC Berkeley 10 July 2014 By the end of this lecture... You will be able to: Use the theory of equally likely

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098%

Coin tosses If a fair coin is tossed 10 times, what will we see? 30% 25% 24.61% 20% 15% 10% Probability 20.51% 20.51% 11.72% 11.72% 5% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% 0 1 2 3 4 5 6 7 8 9 10 Number

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098%

Coin tosses If a fair coin is tossed 10 times, what will we see? 30% 25% 24.61% 20% 15% 10% Probability 20.51% 20.51% 11.72% 11.72% 5% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% 0 1 2 3 4 5 6 7 8 9 10 Number

Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance Free-Response 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is

CSI 23 LECTURE NOTES (Ojakian) Topics 5 and 6: Probability Theory

CSI 23 LECTURE NOTES (Ojakian) Topics 5 and 6: Probability Theory 1. Probability Theory OUTLINE (References: 5.1, 5.2, 6.1, 6.2, 6.3) 2. Compound Events (using Complement, And, Or) 3. Conditional Probability

Midterm 2 Practice Problems

Midterm 2 Practice Problems May 13, 2012 Note that these questions are not intended to form a practice exam. They don t necessarily cover all of the material, or weight the material as I would. They are

3.6 Theoretical and Experimental Coin Tosses

wwwck12org Chapter 3 Introduction to Discrete Random Variables 36 Theoretical and Experimental Coin Tosses Here you ll simulate coin tosses using technology to calculate experimental probability Then you

INTRODUCTORY STATISTICS LECTURE 4 PROBABILITY

INTRODUCTORY STATISTICS LECTURE 4 PROBABILITY THE GREAT SCHLITZ CAMPAIGN 1981 Superbowl Broadcast of a live taste pitting Against key competitor: Michelob Subjects: 100 Michelob drinkers REF: SCHLITZBREWING.COM

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.

Probability of Independent and Dependent Events. CCM2 Unit 6: Probability

Probability of Independent and Dependent Events CCM2 Unit 6: Probability Independent and Dependent Events Independent Events: two events are said to be independent when one event has no affect on the probability

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 Fall 2008 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 3.2 - Measures of Central Tendency

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 Fall 2008 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 3.2 - Measures of Central Tendency

4.3 Rules of Probability

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

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

Name Class Date. Introducing Probability Distributions

Name Class Date Binomial Distributions Extension: Distributions Essential question: What is a probability distribution and how is it displayed? 8-6 CC.9 2.S.MD.5(+) ENGAGE Introducing Distributions Video

Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include your name and student ID.

Math 3201 Unit 3 Probability Test 1 Unit Test Name: Part 1 Selected Response: Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include

Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27

Exercise Sheet 3 jacques@ucsd.edu Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27 1. A six-sided die is tossed.

Independence Is The Word

Problem 1 Simulating Independent Events Describe two different events that are independent. Describe two different events that are not independent. The probability of obtaining a tail with a coin toss

Discrete Random Variables Day 1

Discrete Random Variables Day 1 What is a Random Variable? Every probability problem is equivalent to drawing something from a bag (perhaps more than once) Like Flipping a coin 3 times is equivalent to

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

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

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

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

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as:

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as: E n ( Y) y f( ) µ i i y i The sum is taken over all values

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

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

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

Random Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment.

Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

2 A fair coin is flipped 8 times. What is the probability of getting more heads than tails? A. 1 2 B E. NOTA

For all questions, answer E. "NOTA" means none of the above answers is correct. Calculator use NO calculators will be permitted on any test other than the Statistics topic test. The word "deck" refers

This Probability Packet Belongs to:

This Probability Packet Belongs to: 1 2 Station #1: M & M s 1. What is the sample space of your bag of M&M s? 2. Find the theoretical probability of the M&M s in your bag. Then, place the candy back into

Name: Unit 7 Study Guide 1. Use the spinner to name the color that fits each of the following statements.

1. Use the spinner to name the color that fits each of the following statements. green blue white white blue a. The spinner will land on this color about as often as it lands on white. b. The chance of

SECONDARY 2 Honors ~ Lesson 9.2 Worksheet Intro to Probability

SECONDARY 2 Honors ~ Lesson 9.2 Worksheet Intro to Probability Name Period Write all probabilities as fractions in reduced form! Use the given information to complete problems 1-3. Five students have the

Random Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }

Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count

7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count Probability deals with predicting the outcome of future experiments in a quantitative way. The experiments

EE 126 Fall 2006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO

EE 16 Fall 006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO You have 90 minutes to complete the quiz. Write your solutions in the exam booklet. We will

Probabilities and Probability Distributions

Probabilities and Probability Distributions George H Olson, PhD Doctoral Program in Educational Leadership Appalachian State University May 2012 Contents Basic Probability Theory Independent vs. Dependent

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

Ch Probability Outcomes & Trials

Learning Intentions: Ch. 10.2 Probability Outcomes & Trials Define the basic terms & concepts of probability. Find experimental probabilities. Calculate theoretical probabilities. Vocabulary: Trial: real-world

University of California, Berkeley, Statistics 20, Lecture 1. Michael Lugo, Fall Exam 2. November 3, 2010, 10:10 am - 11:00 am

University of California, Berkeley, Statistics 20, Lecture 1 Michael Lugo, Fall 2010 Exam 2 November 3, 2010, 10:10 am - 11:00 am Name: Signature: Student ID: Section (circle one): 101 (Joyce Chen, TR

A. 15 B. 24 C. 45 D. 54

A spinner is divided into 8 equal sections. Lara spins the spinner 120 times. It lands on purple 30 times. How many more times does Lara need to spin the spinner and have it land on purple for the relative

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 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.

Module 5: Probability and Randomness Practice exercises

Module 5: Probability and Randomness Practice exercises PART 1: Introduction to probability EXAMPLE 1: Classify each of the following statements as an example of exact (theoretical) probability, relative

ITEC 2600 Introduction to Analytical Programming. Instructor: Prof. Z. Yang Office: DB3049

ITEC 2600 Introduction to Analytical Programming Instructor: Prof. Z. Yang Office: DB3049 Lecture Eleven Monte Carlo Simulation Monte Carlo Simulation Monte Carlo simulation is a computerized mathematical

Unit 11 Probability. Round 1 Round 2 Round 3 Round 4

Study Notes 11.1 Intro to Probability Unit 11 Probability Many events can t be predicted with total certainty. The best thing we can do is say how likely they are to happen, using the idea of probability.

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Lecture 4: Data Displays (cont.) Tessa L. Childers-Day UC Berkeley 26 June 2014 By the end of this lecture... You will be able to: Comprehend displays of quantitative

Probability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability

Most people think they understand odds and probability. Do you? Decision 1: Pick a card Decision 2: Switch or don't Outcomes: Make a tree diagram Do you think you understand probability? Probability Write

Date. Probability. Chapter

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

4.12 Practice problems

4. Practice problems In this section we will try to apply the concepts from the previous few sections to solve some problems. Example 4.7. When flipped a coin comes up heads with probability p and tails

c. If you roll the die six times what are your chances of getting at least one d. roll.

1. Find the area under the normal curve: a. To the right of 1.25 (100-78.87)/2=10.565 b. To the left of -0.40 (100-31.08)/2=34.46 c. To the left of 0.80 (100-57.63)/2=21.185 d. Between 0.40 and 1.30 for

Chapter 17: The Expected Value and Standard Error

Chapter 17: The Expected Value and Standard Error Think about drawing 25 times, with replacement, from the box: 0 2 3 4 6 Here s one set of 25 draws: 6 0 4 3 0 2 2 2 0 0 3 2 4 2 2 6 0 6 3 6 3 4 0 6 0,

Chapter 11. Sampling Distributions. BPS - 5th Ed. Chapter 11 1

Chapter 11 Sampling Distributions BPS - 5th Ed. Chapter 11 1 Sampling Terminology Parameter fixed, unknown number that describes the population Statistic known value calculated from a sample a statistic

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

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

Grade 7/8 Math Circles February 25/26, Probability

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely

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

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

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

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

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

Review. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Outline Sec Comparing Rational Numbers

FOUNDATIONS Outline Sec. 3-1 Gallo Name: Date: Review Natural Numbers: Whole Numbers: Integers: Rational Numbers: Comparing Rational Numbers Fractions: A way of representing a division of a whole into

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

Kansas City Area Teachers of Mathematics 2016 KCATM Math Competition STATISTICS and PROBABILITY GRADE 6 INSTRUCTIONS Do not open this booklet until instructed to do so. Time limit: 20 minutes You may use

Appendix E Labs This page intentionally left blank Dice Lab (Worksheet) Objectives: 1. Learn how to calculate basic probabilities of dice. 2. Understand how theoretical probabilities explain experimental

Normal Distribution Lecture Notes Continued

Normal Distribution Lecture Notes Continued 1. Two Outcome Situations Situation: Two outcomes (for against; heads tails; yes no) p = percent in favor q = percent opposed Written as decimals p + q = 1 Why?

Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

Section 8: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as

Waiting Times. Lesson1. Unit UNIT 7 PATTERNS IN CHANCE

Lesson1 Waiting Times Monopoly is a board game that can be played by several players. Movement around the board is determined by rolling a pair of dice. Winning is based on a combination of chance and

Female Height. Height (inches)

Math 111 Normal distribution NAME: Consider the histogram detailing female height. The mean is 6 and the standard deviation is 2.. We will use it to introduce and practice the ideas of normal distributions.

Foundations of Probability Worksheet Pascal

Foundations of Probability Worksheet Pascal The basis of probability theory can be traced back to a small set of major events that set the stage for the development of the field as a branch of mathematics.

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

CCM6+7+ Unit 11 ~ Page 1. Name Teacher: Townsend ESTIMATED ASSESSMENT DATES:

CCM6+7+ Unit 11 ~ Page 1 CCM6+7+ UNIT 11 PROBABILITY Name Teacher: Townsend ESTIMATED ASSESSMENT DATES: Unit 11 Vocabulary List 2 Simple Event Probability 3-7 Expected Outcomes Making Predictions 8-9 Theoretical

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 7.1 Experiments, Sample Spaces, and Events

Section 7.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with observable results. 1. Which of the follow are experiments? (a) Going into a room and turning on a light.

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

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

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

Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability

Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability Lesson Practice Problems Lesson 1: Predicting to Win (Finding Theoretical Probabilities) 1-3 Lesson 2: Choosing Marbles

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

Name. Is the game fair or not? Prove your answer with math. If the game is fair, play it 36 times and record the results.

Homework 5.1C You must complete table. Use math to decide if the game is fair or not. If Period the game is not fair, change the point system to make it fair. Game 1 Circle one: Fair or Not 2 six sided

STAT Chapter 14 From Randomness to Probability

STAT 203 - Chapter 14 From Randomness to Probability This is the topic that started my love affair with statistics, although I should mention that we will only skim the surface of Probability. Let me tell

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,

Ex 1: A coin is flipped. Heads, you win \$1. Tails, you lose \$1. What is the expected value of this game?

AFM Unit 7 Day 5 Notes Expected Value and Fairness Name Date Expected Value: the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities.

CSC/MTH 231 Discrete Structures II Spring, Homework 5

CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the

CCMR Educational Programs

CCMR Educational Programs Title: Date Created: August 6, 2006 Author(s): Appropriate Level: Abstract: Time Requirement: Joan Erickson Should We Count the Beans one at a time? Introductory statistics or

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

Simulations. 1 The Concept

Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that can be

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:

Statistics Laboratory 7

Pass the Pigs TM Statistics 104 - Laboratory 7 On last weeks lab we looked at probabilities associated with outcomes of the game Pass the Pigs TM. This week we will look at random variables associated

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.

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

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

23 Applications of Probability to Combinatorics

November 17, 2017 23 Applications of Probability to Combinatorics William T. Trotter trotter@math.gatech.edu Foreword Disclaimer Many of our examples will deal with games of chance and the notion of gambling.

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

We all know what it means for something to be random. Or do

CHAPTER 11 Understanding Randomness The most decisive conceptual event of twentieth century physics has been the discovery that the world is not deterministic.... A space was cleared for chance. Ian Hocking,

Please Turn Over Page 1 of 7

. Page 1 of 7 ANSWER ALL QUESTIONS Question 1: (25 Marks) A random sample of 35 homeowners was taken from the village Penville and their ages were recorded. 25 31 40 50 62 70 99 75 65 50 41 31 25 26 31

Unit 1B-Modelling with Statistics. By: Niha, Julia, Jankhna, and Prerana

Unit 1B-Modelling with Statistics By: Niha, Julia, Jankhna, and Prerana [ Definitions ] A population is any large collection of objects or individuals, such as Americans, students, or trees about which

Junior Circle Meeting 5 Probability. May 2, ii. In an actual experiment, can one get a different number of heads when flipping a coin 100 times?

Junior Circle Meeting 5 Probability May 2, 2010 1. We have a standard coin with one side that we call heads (H) and one side that we call tails (T). a. Let s say that we flip this coin 100 times. i. How