Thinking Mathematically Expected Value

Expected value is a mathematical way to use probabilities to determine what to expect in various situations over the long run. For example, we can use expected value to find the outcomes of the roll of a fair dice. The outcomes are 1,2,3,4,5, and 6, each with a probability of 1/6. The expected value, E, is computed by multiplying each outcome by its probability and then adding these products. E = 11/6 + 21/6 + 31/6 + 41/6 + 51/6 + 61/6 = ( )/6 = 21/6 = 3.5

Example Computing Expected Value Find the expected value for the number of girls for a family with three children.

Solution A family with three children can have 0,1,2,or 3 girls. There are eight ways these outcomes can occur: No girls: boy boy boy One girl: girl boy boy, boy girl boy, boy boy girl Two girls: girl girl boy, girl boy girl, boy girl girl Three girls: girl girl girl

Solution cont. The expected value, E, is computed by multiplying each outcome by its probability and then adding these products. E = 01/8 + 13/8 + 23/8 + 31/8 = ( )/8 = 12/8 = 3/2 = 1.5 The expected value is 1.5. This means if we record the number of girls in many different three-child families, the average number of girls for all these families will be 1.5 In a three-child family, half the children are expected to be girls, so the expected value is consistent with this observation.

Example Determining an Insurance Premium An automobile insurance company has determined that the probabilities for various claim amounts for drivers ages 16 through 21, shown in the table on the next slide. a.Calculate the expected value and describe what this means in practical terms. b.How much should the company charge an average premium so that it does not lose or gain money on its claim costs?

Calculate the expected value and describe what this means in practical terms. E = $0(0.70) + $2000(0.15) + $4000(0.08) + $6000(0.05) + $8000(0.01) + $10,000(0.01) =$0+$300+$320+$300+$80+$100 =$1100

How much should the company charge an average premium so that it does not lose or gain money on its claim costs? At the very least, the amount that the company should charge as an average premium for each person in the group is $1100. In this way, it will not lose or gain money on its claims costs. It’s quite probable that the company will charge more, moving from break-even to profit.

Example Expected Value as Average Payoff A game is played using one die. If the die is rolled and shows 1, 2, or 3, the player wins nothing. If the die shows 4 or 5, the player wins $3. If the die shows 6, the player wins $9. If there is a charge of $1 to play the game, what is the games expected value? Describe what this means in practical terms.

Solution OutcomesGains or LossProbability 1, 2, or 3-$13/6 4 or 5$22/6 6$81/6 E = (-$1)(3/6) + $2(2/6) + $8(1/6) = (-$3 + $4 + $8)/6 = $9/6 = $1.50 This means in the long run, a player can expect to win an average of $1.50 for each game played.

Example Expected Value and Roulette One way to bet in roulette is to place $1 on a single number. If the ball lands on that number, you are awarded $35 and get to keep the $1 that you paid to play the game. If the ball lands on any one of the other 37 slots, you are awarded nothing and the $1 you bet is collected. Find the expected value for playing roulette if you be $1 on number 20. Describe what this means.

Solution OutcomeGain or LossProbability 20$351/38 Not 20-$137/38 E = $35(1/38) + (-$1)(37/38) = ($35-$37)/38 = -$2/38 = -$0.05 This means that in the long run, a player can expect to lose about 5 cents for each game played.

Thinking Mathematically Expected Value