Chapter 12 Probability 2008 Pearson AddisonWesley All rights

Chapter 12: Probability 12. 1 Basic Concepts 12. 2 Events Involving “Not” and “Or” 12. 3 Conditional Probability; Events Involving “And” 12. 4 Binomial Probability 12. 5 Expected Value 2 © 2008 Pearson Addison-Wesley. All rights reserved

Chapter 1 Section 12 -5 Expected Value © 2008 Pearson Addison-Wesley. All rights reserved

Expected Value • • Expected Value Games and Gambling Investments Business and Insurance 4

Expected Value Children in third grade were surveyed and told to pick the number of hours that they play electronic games each day. The probability distribution is given below. # of Hours x Probability P(x) 0 . 3 1 . 4 2 . 2 3 . 1 5 © 2008 Pearson Addison-Wesley. All rights reserved

Expected Value Compute a “weighted average” by multiplying each possible time value by its probability and then adding the products. 1. 1 hours is the expected value (or the mathematical expectation) of the quantity of time spent playing electronic games. 6 © 2008 Pearson Addison-Wesley. All rights reserved

Expected Value If a random variable x can have any of the values x 1, x 2 , x 3 , …, xn, and the corresponding probabilities of these values occurring are P(x 1), P(x 2), P(x 3), …, P(xn), then the expected value of x is given by 7 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Expected Value Find the expected number of boys for a three-child family. Assume girls and boys are equally likely. Solution S = {ggg, ggb, gbg, bgg, gbb, bgb, bbg, bbb} The probability distribution is on the right. # Boys Probability x P(x) 0 1/8 1 3/8 2 3/8 3 1/8 Product 0 3/8 6/8 3/8 8 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Expected Value Solution (continued) The expected value is the sum of the

Example: Finding Expected Winnings A player pays $3 to play the following game: He rolls a die and receives $7 if he tosses a 6 and $1 for anything else. Find the player’s expected net winnings for the game. 10 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Expected Winnings Solution The information for the game is displayed below. Die Outcome Payoff Net P(x) 1, 2, 3, 4, or 5 $1 –$2 5/6 6 $7 $4 –$10/6 1/6 $4/6 Expected value: E(x) = –$6/6 = –$1. 00 11 © 2008 Pearson Addison-Wesley. All rights reserved

Games and Gambling A game in which the expected net winnings are zero is called a fair game. A game with negative expected winnings is unfair against the player. A game with positive expected net winnings is unfair in favor of the player. 12 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding the Cost for a Fair Game What should the game in the previous example cost so that it is a fair game? Solution Because the cost of $3 resulted in a net loss of $1, we can conclude that the $3 cost was $1 too high. A fair cost to play the game would be $3 – $1 = $2. 13 © 2008 Pearson Addison-Wesley. All rights reserved

Investments Expected value can be a useful tool for evaluating investment opportunities. 14 ©

Example: Expected Investment Profits Mark is going to invest in the stock of one of the two companies below. Based on his research, a $6000 investment could give the following returns. Company ABC Company PDQ Profit or Probability Loss x P(x) –$400. 2 $600. 8 $800. 5 1000. 2 $1300. 3 15 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Expected Investment Profits Find the expected profit (or loss) for each of the two stocks. Solution ABC: –$400(. 2) + $800(. 5) + $1300(. 3) = $710 PDQ: $600(. 8) + $1000(. 2) = $680 16 © 2008 Pearson Addison-Wesley. All rights reserved

Business and Insurance Expected value can be used to help make decisions in various

Example: Expected Lumber Revenue A lumber wholesaler is planning on purchasing a load of lumber. He calculates that the probabilities of reselling the load for $9500, $9000, or $8500 are. 25, . 60, and. 15, respectfully. In order to ensure an expected profit of at least $2500, how much can he afford to pay for the load? 18 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Expected Lumber Revenue Solution The expected revenue from sales can be found below. Income x P(x) $9500 . 25 $2375 $9000 . 60 $5400 $8500 . 15 $1275 Expected revenue: E(x) = $9050 19 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Expected Lumber Revenue Solution (continued) profit = revenue – cost or cost = profit – revenue To have an expected profit of $2500, he can pay up to $9050 – $2500 = $6550. 20 © 2008 Pearson Addison-Wesley. All rights reserved