SECTION 11-5 • Expected Value Slide 11-5-1 EXPECTED VALUE • • • • Expected Value Games and Gambling Investments Business and Insurance Slide 11-5-2 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 Slide 11-5-3 EXPECTED VALUE Compute a “weighted average” by multiplying each possible time value by its probability and then adding the products. 0(.3) 1(.4) 2(.2) 3(.1) 1.1 1.1 hours is the expected value (or the mathematical expectation) of the quantity of time spent playing electronic games. Slide 11-5-4 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 E ( x) x1 P( x1 ) x2 P( x2 ) xn P( xn ). Slide 11-5-5 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 x 0 1 2 3 Probability P(x) 1/8 3/8 3/8 1/8 Product x P( x) 0 3/8 6/8 3/8 Slide 11-5-6 EXAMPLE: FINDING EXPECTED VALUE Solution (continued) The expected value is the sum of the third column: 3 6 3 12 0 8 8 8 8 3 1.5. 2 So the expected number of boys is 1.5. Slide 11-5-7 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. Slide 11-5-8 EXAMPLE: FINDING EXPECTED WINNINGS Solution The information for the game is displayed below. Net P(x) x P( x) $1 –$2 5/6 –$10/6 $7 $4 1/6 $4/6 Die Outcome Payoff 1, 2, 3, 4, or 5 6 Expected value: E(x) = –$6/6 = –$1.00 Slide 11-5-9 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. Slide 11-5-10 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. Slide 11-5-11 INVESTMENTS Expected value can be a useful tool for evaluating investment opportunities. Slide 11-5-12 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 Profit or Probability Loss x P(x) –$400 .2 $800 .5 $1300 .3 Company PDQ Profit or Probability Loss x P(x) $600 .8 1000 .2 Slide 11-5-13 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 Slide 11-5-14 BUSINESS AND INSURANCE Expected value can be used to help make decisions in various areas of business, including insurance. Slide 11-5-15 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? Slide 11-5-16 EXAMPLE: EXPECTED LUMBER REVENUE Solution The expected revenue from sales can be found below. Income x P(x) x P( x) $9500 .25 $2375 $9000 .60 $5400 $8500 .15 $1275 Expected revenue: E(x) = $9050 Slide 11-5-17 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. Slide 11-5-18