1. Chapter 1
Expected Value

2. 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

3. 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

4. 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.

5. 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.

6. 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.

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.

8. 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

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.

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.

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

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

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

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

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?

16. 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

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.