1. 16.6 Expected Value

2. Objective
To find expected value in situations involving gains and losses and to determine whether a game is fair.

3. Payoff x1 x2 x3 … xn Probability P(x1) P(x2) P(x3) P(xn)
If a given situation involves various payoffs then its expected values is calculated as follows.
Payoff x1 x2 x3 … xn
Probability
P(x1)
P(x2)
P(x3)
P(xn)

4. Dice Game Event Die shows 1,2, or 3 Die shows 4 or 5 Die shows 6
Payoff
+10 pts.
-13 pts.
-1 pt.
Probability ⅓
1/6
What is the expected value (payoff)?
.5 point

5. Heads 2 1 Payoff to A $4 -$1 -$2 Probability ¼ ½
Two coins are tossed. If both land heads up, then player A wins $4 from player B. If exactly one coin lands heads up, then B wins $1 from A. If both land tails up then B wins $2 from A .
a) Make a table
Heads 2 1
Payoff to A $4
-$1
-$2
Probability
b) What is the expected value of the game? Is the game fair?
0 game is fair

6. Three cards are drawn at random without replacement, from a standard deck. Find the expected value for the occurrence of hearts.
Make a table
Hearts 1 2 3
Probability
703/1700
741/1700
234/1700
22/1700

7. An 18-year-old student must decide whether to spend $160 for one year’s car collision damage insurance. The insurance carries a $100 deductible which means that the student files a damage claim, the student must pay $110 of the damage amount, with the insurance company paying the rest (up to the value of the car). Because the car is only worth $1500, the student consults with an insurance agent who draws up a table of possible damage amounts and their probabilities based on the driving records for 18-year-olds in the region.

8. Event Accident Costing $1500 Accident Costing $1000
No Accident
Payoff
$1400
$900
$400 $0
Probability
0.05
0.02
0.03
0.9
What is the expected value of the insurance?

9. Expected Payoff = $100
Expected Value = = -$60

10. Example
An investment in Project A will result in a loss of $26,000 with probability 0.30, break even with probability 0.50, or result in a profit of $68,000 with probability An investment in Project B will result in a loss of $71,000 with probability 0.20, break even with probability 0.65, or result in a profit of $143,000 with probability Which investment is better?

11. Tools to calculate E(X)-Project A
Variable (X)- The amount of money received from the investment in Project A
X can assume only x1 , x2 , x3
X= x1 is the event that we have Loss
X= x2 is the event that we are breaking even
X= x3 is the event that we have a Profit
x1=$-26,000
x2=$0
x3=$68,000
P(X= x1)=0.3
P(X= x2)= 0.5
P(X= x3)= 0.2

12. Tools to calculate E(X)-Project B
Variable (X)- The amount of money received from the investment in Project B
X can assume only x1 , x2 , x3
X= x1 is the event that we have Loss
X= x2 is the event that we are breaking even
X= x3 is the event that we have a Profit
x1=$-71,000
x2=$0
x3=$143,000
P(X= x1)=0.2
P(X= x2)= 0.65
P(X= x3)= 0.15

14. Assignment
Page , 17,18,21,22