Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 13.4, Slide 1 13 Probability What Are the Chances?

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 13.4, Slide 2 Expected Value 13.4 Understand the meaning of expected value. Calculate the expected value of lotteries and games of chance. Use expected value to solve applied problems.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 3 Example: The value of several items along with the probabilities that these items will be stolen over the next year are shown. Predict what the Expected Value (continued on next slide) insurance company can expect to pay in claims on your policy. Is $100 a fair premium for this policy?

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 4 Solution: We add an expected payout column to the table. For example for the laptop, $2000 × (.02) = $40. Expected Value (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 5 Solution: We may now compute the expected payout. With an expected payout of $90, a $100 premium is reasonable. Expected Value

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 6 Expected Value

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 7 Example: How many heads we can expect when we flip four fair coins? Expected Value and Games of Chance Solution: There are 16 ways to flip four coins. We could use a tree to complete the table shown. # heads expected

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 8 Example: Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (you also keep your $1 bet); otherwise you lose the $1. What is the expected value of this bet? Solution: This is an experiment with two outcomes: Expected Value and Games of Chance {You win (worth +$35), You lose (worth –$1)}. (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 9 The probability of winning is. The probability of losing is The expected value of the bet is You can expect to lose about 5 cents for every dollar you bet. Expected Value and Games of Chance

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 10 Expected Value and Games of Chance

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 11 Example: Assume that it costs $1 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($499 considering the $1 cost). What is the expected value of this game? Expected Value and Games of Chance (continued on next slide) State Lottery Cost: $1 Prize: $500 $$

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 12 The probability of winning is. The probability of losing is The expected value of this game is You can expect to lose 50 cents for every ticket you buy. Expected Value and Games of Chance

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 13 Example: Assume that it costs $1 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($499 with the $1 cost). What should the price of a ticket be in order to make this game fair? Solution: Let x be the price of a ticket for the lottery to be fair. Then if you win, your profit will be 500 – x and if you lose, your loss will be x. Expected Value and Games of Chance (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 14 The expected value of this game is To be fair, we must have. Expected Value and Games of Chance (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 15 We solve this equation for x. For this game to be fair, a ticket should cost 50 ¢. Expected Value and Games of Chance

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 16 Example: A test consists of multiple-choice questions with five answer choices. One point is earned for each correct answer; point is subtracted for each incorrect answer. Questions left blank neither receive nor lose points. Other Applications of Expected Value (continued on next slide) a) Find the expected value of randomly guessing an answer to a question. Interpret the meaning of this result for the student. b) If you can eliminate one of the choices, is it wise to guess in this situation?

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 17 Solution (a): The probability of guessing correctly is. The probability of guessing incorrectly is. The expected value is You can expect to lose points by guessing. Other Applications of Expected Value

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 18 Solution (b): Eliminating one answer choice then the probability of guessing correctly is. The probability of guessing incorrectly is. The expected value in this case is You now neither benefit nor are penalized by guessing. Other Applications of Expected Value

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 19 Example: The manager of a coffee shop is deciding on how many of bagels to order for tomorrow. According to her records, for the past 10 days the demand has been as follows: She buys bagels for $1.45 each and sells them for $1.85. Unsold bagels are discarded. Find her expected value for her profit or loss if she orders 40 bagels for tomorrow morning. Other Applications of Expected Value (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 20 Solution (a): We must ultimately compute P(demand is 40) × (the profit or loss if demand is 40) + P(demand is 30) × (the profit or loss if demand is 30). The probability that the demand is for 40 bagels is The probability that the demand is for 30 bagels is Other Applications of Expected Value (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 13.4, Slide 21 If demand is 40 bagels: 40($1.85 – $1.45) = 40($0.40) = $16.00 profit If demand is 30 bagels: 30($0.40) = $12.00 profit on bagels sold 10($1.45) = $14.50 loss on bagels not sold Expected profit or loss on 40 bagels is (0.40)(16) + (0.60)(–2.50) = 4.90 profit. Other Applications of Expected Value