16.6 Expected Value

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

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)

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 ½ ⅓ 1/6 What is the expected value (payoff)? .5 point

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 ¼ ½ b) What is the expected value of the game? Is the game fair? 0 game is fair

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

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.

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?

Expected Payoff = $100 Expected Value = 100 -160 = -$60

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 0.20. 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 0.15. Which investment is better?

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

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

Assignment Page 633 1-14, 17,18,21,22