1. 13.4 Expected Value Understand the meaning of expected value. Understand the meaning of expected value. Use expected value to solve applied problems. Use expected value to solve applied problems. Calculate the expected value of lotteries and games of chance. Calculate the expected value of lotteries and games of chance.

2. Assume that an experiment has outcomes numbered 1 to n with probabilities P 1, P 2, P 3, …, P n. Assume that each outcome has a numerical value associated with it and these are labeled V 1, V 2, V 3, …, V n. The expected value of the experiment is (P 1 V 1 ) + (P 2 V 2 ) + (P 3 V 3 ) + … + (P n V n ). Assume that an experiment has outcomes numbered 1 to n with probabilities P 1, P 2, P 3, …, P n. Assume that each outcome has a numerical value associated with it and these are labeled V 1, V 2, V 3, …, V n. The expected value of the experiment is (P 1 V 1 ) + (P 2 V 2 ) + (P 3 V 3 ) + … + (P n V n ). –A game that has an expected value of O is called a fair game. A game in which the expected value is not 0 is an unfair game.

3. Roulette

4. There are many ways to bet on the 38 numbers of a roulette wheel. One simple betting scheme is to place a $1 bet on a single number. If you win, the casino pays you $35 (you also keep your $1 bet). If not, you lose the $1. What is the expected value of this bet? There are many ways to bet on the 38 numbers of a roulette wheel. One simple betting scheme is to place a $1 bet on a single number. If you win, the casino pays you $35 (you also keep your $1 bet). If not, you lose the $1. What is the expected value of this bet?

5. Solution There are 38 likely numbers that can occur. There are 38 likely numbers that can occur. Your number comes up and the value to you is +$35. The probability of this is 1/38. Your number comes up and the value to you is +$35. The probability of this is 1/38. Your number doesn’t come up and the value to you is -$1. The probability of this is 37/38. Your number doesn’t come up and the value to you is -$1. The probability of this is 37/38. The expected value of this bet is: (1/38 · 35) + (37/38 · -1) = -0.0526 The expected value of this bet is: (1/38 · 35) + (37/38 · -1) = -0.0526 On average, the casino expects you to lose slightly more that 5 cents for every dollar you bet. On average, the casino expects you to lose slightly more that 5 cents for every dollar you bet.

6. Classwork/Homework Classwork – Page 766 (5 - 9 odd, 21, 23) Classwork – Page 766 (5 - 9 odd, 21, 23) Homework – Page 766 (6 – 10 even, 22, 24) Homework – Page 766 (6 – 10 even, 22, 24)

7. Dice Table 123456 1(1,1)(1,2)(1,3)(1,4)(1,5)(1,6) 2(2,1)(2,2)(2,3)(2,4)(2,5)(2,6) 3(3,1)(3,2)(3,3)(3,4)(3,5)(3,6) 4(4,1)(4,2)(4,3)(4,4)(4,5)(4,6) 5(5,1)(5,2)(5,3)(5,4)(5,5)(5,6) 6(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)