1. Section 12.4 Expected Value (Expectation)

2. What You Will Learn
Expected Value
Fair Price

3. Expected Value
Expected value, also called expectation, is often used to determine the expected results of an experiment or business venture over the long term.

4. Expected Value
The symbol P1 represents the probability that the first event will occur, and A1 represents the net amount won or lost if the first event occurs.
P2 is the probability of the second event, and A2 is the net amount won or lost if the second event occurs.
And so on…

5. Example 1: A Building Contract
The Rich Walker Construction Company has just received a building contract. From past experience, Rich estimates there is a 60% chance of making a $500,000 profit, a 10% chance of breaking even, and a 30% chance of losing $200,000, depending on weather conditions and other factors. What is the expected value of the company’s contract?

6. Example 1: A Building Contract
Solution
Probability of a gain of $500,000 is 60%, of breaking even is 10%, and a loss of $200,000 is 30%.
Rich’s expectation
= P1A1 + P2A2 + P3A3
= (0.6)($500,000) + (0.1)($0) (0.3)(–$200,000)
= $240,000

7. Example 1: A Building Contract
Solution
Rich has an expectation, or expected average gain, of $240,000 for this contract. Thus, if the company receives similar contracts like this one, with these particular probabilities and amounts, in the long run the company would have an average gain of $240,000 per contract.

8. Example 1: A Building Contract
Solution
However, you must remember that there is a 30% chance the company will lose $200,000 on this particular contract or any particular contract with these probabilities and amounts.

9. Example 5: Winning a Door Prize
When Josh Rosenberg attends a charity event, he is given the opportunity to purchase a ticket for the $50 door prize. The cost of the ticket is $2, and 100 tickets will be sold. Determine Josh’s expectation if he purchases one ticket.

10. Example 5: Winning a Door Prize
Solution
P(Win) = 1/100, net winnings = $48
P(not Win) = 99/100, loss = $2
Expectation =
P(win) • amt win + P(lose) • amt lose
Josh’s expectation is –$1.50 when he purchases one ticket.

11. Fair Price
The fair price is the amount to be paid that will result in an expected value of $0. The fair price may be found by adding the cost to play to the expected value.
Fair price = expected value + cost to play

12. Example 7: Expectation and Fair Price
At a game of chance, the expected value is found to be –$1.50, and the cost to play the game is $4.00. Determine the fair price to play the game.

13. Example 7: Expectation and Fair Price
Solution
Fair price = expected value + cost to play
= – = 2.50
The fair price to play this game would be $2.50. If the cost to play was $2.50 instead of $4.00, the expected value would be $0.