Warm Up What is the type of sampling for 1-3? 1. “The names of 50 contestants are written on 50 cards. The cards are placed in a hat and 10 names are drawn.”

Slides:



Advertisements
Similar presentations
Theoretical Probability
Advertisements

A measurement of fairness game 1: A box contains 1red marble and 3 black marbles. Blindfolded, you select one marble. If you select the red marble, you.
Random Variables and Expectation. Random Variables A random variable X is a mapping from a sample space S to a target set T, usually N or R. Example:
Games of probability What are my chances?. Roll a single die (6 faces). –What is the probability of each number showing on top? Activity 1: Simple probability:
Horse race © Horse race: rules 1.Each player chooses a horse and puts it into a stall. Write your name next to the.
Random Variables A Random Variable assigns a numerical value to all possible outcomes of a random experiment We do not consider the actual events but we.
Chapter 7 Expectation 7.1 Mathematical expectation.
Fair Games/Expected Value
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. Understanding Probability and Long-Term Expectations Chapter 16.
Warm up: Solve each system (any method). W-up 11/4 1) Cars are being produced by two factories, factory 1 produces twice as many cars (better management)
Section 5.1 What is Probability? 5.1 / 1. Probability Probability is a numerical measurement of likelihood of an event. The probability of any event is.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. Understanding Probability and Long-Term Expectations Chapter 16.
1.3 Simulations and Experimental Probability (Textbook Section 4.1)
Probability Evaluation 11/12 th Grade Statistics Fair Games Random Number Generator Probable Outcomes Resources Why Fair Games? Probable Outcome Examples.
5.1 Probability in our Daily Lives.  Which of these list is a “random” list of results when flipping a fair coin 10 times?  A) T H T H T H T H T H 
Warm Up If Babe Ruth has a 57% chance of hitting a home run every time he is at bat, run a simulation to find out his chances of hitting a homerun at least.
Warm Up Section Explain why the following is not a valid probability distribution? 2.Is the variable x discrete or continuous? Explain. 3.Find the.
Chapter 16 Week 6, Monday. Random Variables “A numeric value that is based on the outcome of a random event” Example 1: Let the random variable X be defined.
Bernoulli Trials, Geometric and Binomial Probability models.
Introduction to Probability – Experimental Probability.
Section 6.2: Probability Models Ways to show a sample space of outcomes of multiple actions/tasks: (example: flipping a coin and rolling a 6 sided die)
The Mean of a Discrete Random Variable Lesson
Unit 4 Section 3.1.
By:Tehya Pugh. What is Theoretical Probability  Theoretical Probability Is what you predict what will happen without really doing the experiment.  I.
When could two experimental probabilities be equal? Question of the day.
Probability Distributions. Constructing a Probability Distribution Definition: Consists of the values a random variable can assume and the corresponding.
In games of chance the expectations can be thought of as the average outcome if the game was repeated multiple times. Expectation These calculated expectations.
16.6 Expected Value.
Copyright © Cengage Learning. All rights reserved. Probability and Statistics.
Welcome to MDM4U (Mathematics of Data Management, University Preparation)
Copyright © 2009 Pearson Education, Inc.
Probability the likelihood of specific events
Chapter 11 Probability.
Sec. 4-5: Applying Ratios to Probability
Chapter 6 6.1/6.2 Probability Probability is the branch of mathematics that describes the pattern of chance outcomes.
Copyright © Cengage Learning. All rights reserved.
Honors Stats 4 Day 10 Chapter 16.
Common Core Math III Unit 1: Statistics
Determining the theoretical probability of an event
(Single and combined Events)
Discrete Distributions
Stats 4 Day 20b.
CHAPTER 6 Random Variables
Expected Value.
Probability: Living with the Odds
Warm Up 4.
Chapter 17 Thinking about Chance.
Expected Value and Fair Games
Advanced Placement Statistics Section 7.2: Means & Variance of a
Probability Trees By Anthony Stones.
The Law of Large Numbers
Chapter 6: Random Variables
Probability Key Questions
Expected Value.
Chapter 14 – From Randomness to Probability
Random Variable Random Variable – Numerical Result Determined by the Outcome of a Probability Experiment. Ex1: Roll a Die X = # of Spots X | 1.
Chapter 3.1 Probability Students will learn several ways to model situations involving probability, such as tree diagrams and area models. They will.
Warm Up Problem of the Day Lesson Presentation Lesson Quizzes.
Probability.
Using Probabilities to Make Fair Decisions
Chapter 1 Expected Value.
Expected Value.
Gain Expected Gain means who much – on average – you would expect to win if you played a game. Example: Mark plays a game (for free). He throws a coin.
Probability True or False?.
Fun… Tree Diagrams… Probability.
Probability in Baseball
Using Probabilities to Make Fair Decisions
Probability.
Homework Due Tomorrow mrsfhill.weebly.com.
Chapter 11 Probability.
Presentation transcript:

Warm Up What is the type of sampling for 1-3? 1. “The names of 50 contestants are written on 50 cards. The cards are placed in a hat and 10 names are drawn.” 2. A researcher selects 5 males and 5 females out of 50 males and 50 females. 3. What would you have to score to be in the 85th percentile if the average score is a 75 and the standard deviation is 4.7? 4. What would be the minimum and maximum value of flipping a coin when using randInt(

Review Simulation: If Babe Ruth has a 57% chance of hitting a home run every time he is at bat, run a simulation to find out his chances of hitting a homerun 3 times if he get to bat 4 times in a game. Run 15 trials.

Day 8: Expected Value Unit 1: Statistics

Today’s Objectives Students will work with expected value.

Expected Value and Fair Games

Expected Value Expected value is the weighted average of all possible outcomes. For example, if a trial has the outcomes 10, 20 and 60: The average of 10, 20, and 60 = 30 This assumes an even distribution: 10 20    60

Sometimes, outcomes will not have equal likelihoods. X 1 2 3 P(X) .5 .25 E(X) = .5(1) + .25(2) + .25(3) = 1.75

You play a game in which you roll one fair die You play a game in which you roll one fair die. If you roll a 6, you win $5. If you roll a 1 or a 2, you win $2. If you roll anything else, you don’t win any money.   Create a probability model for this game:   6  1, 2  3, 4, 5  X  $5  $2  $0  P(X)  1/6  1/3  1/2  How much would you be willing to pay to play?   E(X) =$5(1/6) + $2(1/3) + $0(1/2) = $1.50   A price of $1.50 makes this a fair game.  

E(X) = .05(3500) + .1(2500) + .25(500)+.6(-1000) = -$50. At Tucson Raceway Park, your horse, My Little Pony, has a probability of 1/20 of coming in first place, a probability of 1/10 of coming in second place, and a probability of ¼ of coming in third place. First place pays $4,500 to the winner, second place $3,500 and third place $1,500. Is it worthwhile to enter the race if it costs $1,000? 1st 2nd 3rd Other X $3500 $2500 $500 -$1000 P(X) .05 .10 .25 .60 E(X) = .05(3500) + .1(2500) + .25(500)+.6(-1000) = -$50.

This is the Law of Large Numbers! What does an expected value of -$50 mean? Its important to note that nobody will actually lose $50—this is not one of the options. Over a large number of trials, this will be the average loss experienced. This is the Law of Large Numbers! Insurance companies and casinos build their businesses based on the law of large numbers.

At a carnival there is a game where you get to flip a coin twice At a carnival there is a game where you get to flip a coin twice. For every head they give you $3, and for every tail you have to pay $1. What should the cost of the game be to make it fair?

Questions about expected value?