SECTION 11-5 Expected Value Slide

EXPECTED VALUE Expected Value Games and Gambling Investments Business and Insurance Slide

EXPECTED VALUE Slide 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)

EXPECTED VALUE Slide 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.

EXPECTED VALUE Slide 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

EXAMPLE: FINDING EXPECTED VALUE Slide 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.

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

EXAMPLE: FINDING EXPECTED WINNINGS Slide 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.

EXAMPLE: FINDING EXPECTED WINNINGS Slide 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

GAMES AND GAMBLING Slide 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.

EXAMPLE: FINDING THE COST FOR A FAIR GAME Slide 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.

INVESTMENTS Slide Expected value can be a useful tool for evaluating investment opportunities.

EXAMPLE: EXPECTED INVESTMENT PROFITS Slide Company ABCCompany PDQ Profit or Loss x Probability P(x) Profit or Loss x Probability P(x) –$400.2$600.8 $ $ 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.

EXAMPLE: EXPECTED INVESTMENT PROFITS Slide 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.

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

EXAMPLE: EXPECTED LUMBER REVENUE Slide 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?

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

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