1. SECTION 11-5 Expected Value Slide 11-5-1

2. EXPECTED VALUE Expected Value Games and Gambling Investments Business and Insurance Slide 11-5-2

3. EXPECTED VALUE Slide 11-5-3 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 xProbability P(x) 0.3 1.4 2.2 3.1

4. EXPECTED VALUE Slide 11-5-4 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.

5. EXPECTED VALUE Slide 11-5-5 If a random variable x can have any of the values x 1, x 2, x 3,…, x n, and the corresponding probabilities of these values occurring are P(x 1 ), P(x 2 ), P(x 3 ), …, P(x n ), then the expected value of x is given by

6. EXAMPLE: FINDING EXPECTED VALUE Slide 11-5-6 Find the expected number of boys for a three-child family. Assume girls and boys are equally likely. Solution # BoysProbabilityProduct xP(x)P(x) 01/80 13/8 2 6/8 31/83/8 S = {ggg, ggb, gbg, bgg, gbb, bgb, bbg, bbb} The probability distribution is on the right.

7. EXAMPLE: FINDING EXPECTED VALUE Slide 11-5-7 Solution (continued) The expected value is the sum of the third column: So the expected number of boys is 1.5.

8. EXAMPLE: FINDING EXPECTED WINNINGS Slide 11-5-8 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.

9. EXAMPLE: FINDING EXPECTED WINNINGS Slide 11-5-9 Die OutcomePayoff NetP(x)P(x) 1, 2, 3, 4, or 5$1–$25/6–$10/6 6$7$41/6$4/6 Solution The information for the game is displayed below. Expected value: E(x) = –$6/6 = –$1.00

10. GAMES AND GAMBLING Slide 11-5-10 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.

11. EXAMPLE: FINDING THE COST FOR A FAIR GAME Slide 11-5-11 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.

12. INVESTMENTS Slide 11-5-12 Expected value can be a useful tool for evaluating investment opportunities.

13. EXAMPLE: EXPECTED INVESTMENT PROFITS Slide 11-5-13 Company ABCCompany PDQ Profit or Loss x Probability P(x) Profit or Loss x Probability P(x) –$400.2$600.8 $800.51000.2 $1300.3 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.

14. EXAMPLE: EXPECTED INVESTMENT PROFITS Slide 11-5-14 Solution ABC: –$400(.2) + $800(.5) + $1300(.3) = $710 PDQ: $600(.8) + $1000(.2) = $680 Find the expected profit (or loss) for each of the two stocks.

15. BUSINESS AND INSURANCE Slide 11-5-15 Expected value can be used to help make decisions in various areas of business, including insurance.

16. EXAMPLE: EXPECTED LUMBER REVENUE Slide 11-5-16 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?

17. EXAMPLE: EXPECTED LUMBER REVENUE Slide 11-5-17 Income xP(x)P(x) $9500.25$2375 $9000.60$5400 $8500.15$1275 Solution The expected revenue from sales can be found below. Expected revenue: E(x) = $9050

18. EXAMPLE: EXPECTED LUMBER REVENUE Slide 11-5-18 Solution (continued) profit = revenue – cost or cost = profit – revenue To have an expected profit of $2500, he can pay up to $9050 – $2500 = $6550.