Chapter 1 Expected Value

Expected Value If a random variable x can have any of the values x1, x2 , x3 ,…, xn, and the corresponding probabilities of these values occurring are P(x1), P(x2), P(x3), …, P(xn), then the expected value of x is given by

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) .3 1 .4 2 .2 3 .1

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 of the quantity of time spent playing electronic games.

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

Example: Finding Expected Value 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 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 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 –$10/6 6 $7 $4 1/6 $4/6 Expected value: E(x) = –$6/6 = –$1.00

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.

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.

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

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 Loss x Probability P(x) –$400 .2 $600 .8 $800 .5 1000 $1300 .3

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

Business and Insurance Expected value can be used to help make decisions in various areas of business, including insurance.

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?

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

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.