Chapter 4 Probability Statistics Section 4 3 Expected

Chapter 4: Probability & Statistics Section 4. 3: Expected Value & Fair Price APPLIED MATHEMATICS, FOURTH EDITION, MATOVINA & YATES

Expected Value EV = P(win) × ($ won) + P(lose) × ($ lost) APPLIED MATHEMATICS, FOURTH EDITION, MATOVINA & YATES

Fair Price A game is considered to be fair when the expected value is equal to zero. The fair price to pay, FP, for playing a game can be found by calculating the expected winnings. FP = P(win) × ($won) Example: A carnival game has a player reach into a paper bag and select one of four bills at random. In the bag are two $1 bills, a $10 bill and a $100 bill. What is the fair price to pay for playing this game? Would you play this game if it cost $20 to play? FP = P(Drawing $1) × $1 + P(Drawing $10) × $10 + P(Drawing $100) × $100 FP = (2/4)($1) + (1/4)($100) = $112/4 = $28 If it costs $20, play it. The player has the advantage for ANY cost to play less than $28. APPLIED MATHEMATICS, FOURTH EDITION, MATOVINA & YATES

From a Different Point of View The Expected Value (EV) and Fair Price (FP) calculations are very similar. In fact, • EV = FP – Cost Think of the raffle ticket example in which 500 tickets were sold at a cost of $2 each, and the single prize was $750. • EV = FP – Cost • EV = (1/500)($750) - $2 = $1. 50 - $2. 00 = -$0. 50 Also keep in mind: • If the cost is equal to the FP, the EV = 0, which makes it a fair game. • If the cost is greater than the FP, the EV is negative from the player’s perspective. • If the cost is less than the FP, the EV is positive from the player’s perspective. APPLIED MATHEMATICS, FOURTH EDITION, MATOVINA & YATES