Section 12.4 Expected Value (Expectation)

What You Will Learn Expected Value Fair Price

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.

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…

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?

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

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.

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.

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.

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.

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

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.

Example 7: Expectation and Fair Price Solution Fair price = expected value + cost to play = –1.50 + 4.00 = 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.