answer question circled 426 PART V From the Data at Hand to the World at Large 4

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answer question circled

426 PART V From the Data at Hand to the World at Large 4. New game You pay $10 and roll a die. If you get a 6, you win $50. If not, you get to roll again. If you get a 6 this time, you get your $10 back. a) Create a probability model for this game. b) Find the expected value and standard deviation of your prospective winnings. c) You play this game five times. Find the expected value and standard deviation of your average winnings. d) 100 people play this game. What's the probability the person running the game makes a profit? 55. AP Stats 2011 The College Board reported the score dis- tribution shown in the table for all students who took the 2011 AP Statistics exam. Percent of Students 12.4 21.4 25.1 17.6 23.5 Score a) Find the mean and standard deviation of the scores.

Explanation / Answer

a) X = -10, 0 or 40 for your winnings (or losses). The corresponding probabilities are:

P(40) = 1/6
P(0) = (5/6)(1/6) = 5/36
P(-10) = 1 - 1/6 - 5/36 = 25/36

b) E(X) = (-10)(25/36) + 0(5/36) + 40(1/6) = -5/18

E(X^2) = (100)(25/36) + 1600(1/6) = 3025 / 9

Var(X) = E(X^2) - [E(X)]^2 = 3025 / 9 - (-5/18)^2 = 108875 / 324

Std Dev. = sqrt [Var X] = 119043 / 6494 equal approximately 18.331

c) E(X) = (-10)(25/36) + 0(5/36) + 40(1/6) = -5/18

5 * E(X) = - 25/18

Std Dev. = 5 * 18.331 = 91.65

d) Probability that customer loses:

P(X < 0) = P{ z < [0 - (-5/18)] / [(119043 / 6494) / sqrt 100] } = P(z < 0.1515) = 0.560

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