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problems11-sol - MATH S-104 Lecture 11 In-class Problem...

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Unformatted text preview: MATH S-104 Lecture 11 In-class Problem Solutions July 28, 2009 Problem 1 ‡ : In the original game of Carnival Dice, the player chooses a number from 1 to 6. She then throws three fair and mutually independent dice. She wins one dollar if any die matches, and loses a dollar otherwise. This is a losing proposition for the player. Consider a modified version of Carnival Dice. The game is the same except the player wins one dollar for each die that matches her number, and she loses one dollar if no die matches. a) Let M be a random variable that is the number of matches. What is p ( M = n ) ? The probability that n dice match is P ( M = n ) = 3 n ⋅ 5 6 3- n 1 6 n b) Is this a good game to play? What is the expected profit? Let X be the profit. The expected profit is E ( X ) =- 1 ⋅ p ( M = ) + 1 ⋅ p ( M = 1 ) + 2 ⋅ p ( M = 2 ) + 3 ⋅ p ( M = 3 ) . Plugging in the probability of each possible number of matches, we get E ( X ) =- 1 ⋅ 5 6 3 + 1 ⋅ 3 ⋅ 1 6 ⋅ 5 6 2 + 2 ⋅ 3 ⋅...
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