Chapter 14 sec 4 Expected Value. Questions How many of you have car insurance? How many of you have health insurance? Do you wonder how the insurance.

Slides:

Advertisements

Similar presentations

Expected Value. When faced with uncertainties, decisions are usually not based solely on probabilities A building contractor has to decide whether to.

Advertisements

Section 6.3 ~ Probabilities With Large Numbers

Utility Theory.

Math 409/409G History of Mathematics Expected Value.

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.

Clear your desk for your quiz. Unit 2 Day 8 Expected Value Average expectation per game if the game is played many times Can be used to evaluate and.

Take out a coin! You win 4 dollars for heads, and lose 2 dollars for tails.

Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 13.4, Slide 1 13 Probability What Are the Chances?

Warm up 1)We are drawing a single card from a standard deck of 52 find the probability of P(seven/nonface card) 2)Assume that we roll two dice and a total.

DGame Task Task 1Task 2Task 3Task 4 Task 5Task 6Task 7Task 8 Task 9Task 10Task 11 NC Level 3 to 8.

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:

Mathematical Expectation. A friend offers you the chance to play the following game: You bet $2 and roll a die. If you roll a 6 you win $5 plus your bet.

1 Many people debate basic questions of chance in games such as lotteries. The Monty Hall problem is a fun brain teaser that Marilyn vos Savant addressed.

Lectures in Microeconomics-Charles W. Upton Minimax Strategies.

Probability and Expected Value 6.2 and 6.3. Expressing Probability The probability of an event is always between 0 and 1, inclusive. A fair coin is tossed.

Expected Value.  In gambling on an uncertain future, knowing the odds is only part of the story!  Example: I flip a fair coin. If it lands HEADS, you.

Minimax Strategies. Everyone who has studied a game like poker knows the importance of mixing strategies. –With a bad hand, you often fold –But you must.

Using Tinkerplots Ruth Kaniuk Endeavour Teacher Fellow, 2013.

4.2 (cont.) Expected Value of a Discrete Random Variable A measure of the “middle” of the values of a random variable.

Fair Games/Expected Value

Probabilities of common Games How do I avoid bad bets?

Quiz 4  Probability Distributions. 1. In families of three children what is the mean number of girls (assuming P(girl)=0.500)? a) 1 b) 1.5 c) 2 d) 2.5.

Warm-up The mean grade on a standardized test is 88 with a standard deviation of 3.4. If the test scores are normally distributed, what is the probability.

Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 13.4, Slide 1 13 Probability What Are the Chances?

Honors Precalculus: Do Now 1.) You are betting with your friend. Every time you flip heads, your friend pays you a dollar. Every time you flip tails, you.

Section 8.2: Expected Values

Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. Understanding Probability and Long-Term Expectations Chapter 16.

Section 11.6 Odds and Expectation Math in Our World.

Please turn off cell phones, pagers, etc. The lecture will begin shortly.

1 Introduction to Discrete Probability Rosen, Section 6.1 Based on slides by Aaron Bloomfield and …

TIMES 3 Technological Integrations in Mathematical Environments and Studies Jacksonville State University 2010.

1 MAT116 Chapter 4: Expected Value : Summation Notation  Suppose you want to add up a bunch of probabilities for events E 1, E 2, E 3, … E 100.

Produced by MEI on behalf of OCR © OCR 2013 Introduction to Quantitative methods Probability and risk © OCR 2014.

Expected Value MATH 102 Contemporary Math S. Rook.

© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 11 Counting Methods and Probability Theory.

Section 13.4 Expected Value. Objectives 1. Compute expected value. 2. Use expected value to solve applied problems. 3. Use expected value to determine.

1 M14 Expected Value, Discrete  Department of ISM, University of Alabama, ’95,2002 Lesson Objectives  Understand the meaning of “expected value.” (Know.

Outline Random processes Random variables Probability histograms

Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds Discussion Paragraph 7B 1 web 59. Lottery Chances 60. HIV Probabilities 1 world.

MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can.

THE NATURE OF PROBABILITY Copyright © Cengage Learning. All rights reserved. 13.

13.4 Expected Value Understand the meaning of expected value. Understand the meaning of expected value. Use expected value to solve applied problems. Use.

Expected Value.

Statistics Probability Distributions – Part 1. Warm-up Suppose a student is totally unprepared for a five question true or false test and has to guess.

Probability Evaluation 11/12 th Grade Statistics Fair Games Random Number Generator Probable Outcomes Resources Why Fair Games? Probable Outcome Examples.

CHAPTER 16 Expected Value of a Discrete Random Variable A measure of the “middle” of the values of a random variable.

Not So Great Expectations! Which game do you think has the highest return per dollar?

MAT 1000 Mathematics in Today's World. Last Time.

WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6.

L56 – Discrete Random Variables, Distributions & Expected Values

1 Mathematical Expectation Mathematical Expectation Ernesto Diaz, Mathematics Department Redwood High School.

EXPECTATION “EXPECTED VALUE”. E XPECTATION – TELLS YOU THE AVERAGE AMOUNT OF MONEY YOU SHOULD EXPECT TO WIN OR LOSE IN A LOTTERY. PA Daily number: Play:

The Mean of a Discrete Random Variable Lesson

Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds Discussion Paragraph 7B 1 web 59. Lottery Chances 60. HIV Probabilities 1 world.

Thinking Mathematically Expected Value. Expected value is a mathematical way to use probabilities to determine what to expect in various situations over.

Copyright © 2009 Pearson Education, Inc. 6.3 Probabilities with Large Numbers LEARNING GOAL Understand the law of large numbers, use this law to understand.

Add Subtopics Here Games of Chance. Introduction  One of the keys to responsible gambling is an understanding of the mathematics governing the operation.

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.

Ch 11.7 Probability. Definitions Experiment – any happening for which the result is uncertain Experiment – any happening for which the result is uncertain.

Probability in gambling Lucie Tejessyová. Gambling Games of chances or skills when betting ratios are to bettors disadvantage The best bets – bring the.

Lesson 96 – Expected Value & Variance of Discrete Random Variables HL2 Math - Santowski.

Copyright © Cengage Learning. All rights reserved. Probability and Statistics.

4.2 Random Variables and Their Probability distributions

Copyright © Cengage Learning. All rights reserved.

4.2 (cont.) Expected Value of a Discrete Random Variable

Expected Value.

Expected Value.

Counting Methods and Probability Theory

Expected Value.

Counting Methods and Probability Theory

Presentation transcript:

Chapter 14 sec 4 Expected Value

Questions How many of you have car insurance? How many of you have health insurance? Do you wonder how the insurance companies make their money?

It’s a math game. For example, you buy car insurance. Why?

You are playing a sort of mathematical game with the car insurance company. You are betting that you are going to have an accident in the future. While the insurance company is betting that you won’t.

How about the casinos? They are relying on this same mathematical theory-EXPECTED VALUE.

Expected Value Uses probability to compare alternatives to help us make decision.

Definition Assume that an experiment has outcomes numbered 1 to n with probabilities P 1, P 2, P 3, …,P n. Assume that each outcome has a numerical value associated with it and these are labeled V 1, V 2, V 3, …, V n. The expected value of the experiment is

Computing expected value when flipping coins What is the number of heads we can expect when we flip four fair coins?

Remember there are 16 ways to flip four coins. Here is a table.

Expected value of a Roulette wheel There are many ways to bet on the 38 numbers of a roulette wheel, one sample betting scheme is to place a bet, $1, on a single number. In this case, the casino pays you $35( you keep your $1 bet) if your number comes up otherwise you lose the $1.

What is the expected value? 1. Your number comes up and the value to you is +$ Your number doesn’t come up and the value to you is -$1.

Since there are 38 equally likely numbers that can occur, the probability of the first outcome is and the probability of the second outcome is.

The expected value of this bet is

Therefore, this game is unfair game.

Def of fair and unfair game Fair- ▫If a game has an expected value of 0. Unfair- ▫A game in which the expected value is not 0

Assume that it cost $1 to play a state’s daily number. The player chooses a three-digit number between 000 and 999 inclusive, and if the number is selected that day, then the player wins $500(this means the player’s profit is $500-$1 =$499.) A) What is the expected value of this game? B) What should the price of a ticket be in order to make this game unfair?

A) There is 1000 possible numbers that can be selected. Therefore the probability of your winnings is The probability of you losing is

Expected value What does this mean?

This means that the player, on average, can expect to lose 50 cents per game.

B) Let x be the price of a ticket for the lottery to be fair. Then if you win, your profit will be 500-x and if you lose, your loss will be x. We will then recalculate the expected value. We want the game to be fair, so we will set this expected value equal to zero.

This means that 50 cents would be a fair price for a ticket to play this lottery. But if this happens then the state will not make money.

Lotteries Did you know that lotteries existed since ancient times. The Roman emperor Nero gave slaves or villas as door prizes to guests attending his banquets. Augustus Caesar used public lotteries to raise funds to repair Rome. The first public lottery paying money prizes began in Florence, Italy, in the 1500s.

In 1776, the Continental Congress used a lottery to raise $10 million to finance the American Revolution