Probability

1. Probability What Are the Chances?

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.

3. Expected Value • 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 insurance company can expect to pay in claims on your policy. Is $100 a fair premium for this policy? (continued on next slide)

4. Expected Value • Solution: We add an expected payout column to the table. For example for the laptop, $2000 × (.02) = $40. (continued on next slide)

5. Expected Value • Solution: We may now compute the expected payout. • With an expected payout of $90, a $100 premium is reasonable.

6. Expected Value

7. Expected Value and Games of Chance • Example: How many heads we can expect when we flip four fair coins?

8. Expected Value and Games of Chance • Example: How many heads we can expect when we flip four fair coins? • Solution: There are 16 ways to flip four coins. We could use a tree to complete the table shown. # heads expected

9. Expected Value and Games of Chance • 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? (continued on next slide)

10. Expected Value and Games of Chance • 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: {You win (worth +$35), You lose (worth –$1)}. (continued on next slide)

11. Expected Value and Games of Chance 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.

12. Expected Value and Games of Chance

13. State Lottery Cost: $1 Prize: $500 $ $ Expected Value and Games of Chance • 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? (continued on next slide)

14. Expected Value and Games of Chance 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.

15. Expected Value and Games of Chance • 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? (continued on next slide)

16. Expected Value and Games of Chance • 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 xbe the price of a ticket for the lottery to be fair. Then if you win, your profit will be 500 – xand if you lose, your loss will be x. (continued on next slide)

17. Expected Value and Games of Chance The expected value of this game is To be fair, we must have . (continued on next slide)

18. Expected Value and Games of Chance We solve this equation for x. For this game to be fair, a ticket should cost 50¢.

19. Other Applications of Expected Value • 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. 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? (continued on next slide)

20. Other Applications of Expected Value • 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.

21. Other Applications of Expected Value • 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.

22. Other Applications of Expected Value • 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. (continued on next slide)

23. Other Applications of Expected Value • 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 (continued on next slide)

24. Other Applications of Expected Value 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.