homework5_solutions - Math 1011 Intro to Statistics Spring...

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Math 1011– Intro to Statistics Spring 2011 HOMEWORK 5 Due Friday, 3/4/11 1. In roulette, a ball is spun on a wheel and lands one one of the numbers 0, 00, or 1 through 36 (38 total options). There are many different kinds of bets one can make. Find the expected value and standard error for each of the following $1 bets: (a) A bet on a single number pays 35 to 1 (meaning, if you bet $1 and win, you get back your $1 plus $35). (b) A bet on 1-18 pays 1 to 1. (c) A bet on 1-12 pays 2 to 1. (d) A bet on one of 0,00,1,2, or 3 pays 6 to 1. Solution: (a) Let X be 35 if it lands on your number, - 1 otherwise. Then E ( X ) = 35( 1 38 )+( - 1)( 37 38 ) = - 1 19 ≈ - 0 . 053, SE ( X ) = radicalBig (35 + 1 19 ) 2 1 38 + ( - 1 + 1 19 ) 2 37 38 5 . 76 (b) Let X be 1 if it lands on 1-18, - 1 otherwise. E ( X ) = 1( 18 38 )+( - 1) 20 38 = - 1 19 , SE ( X ) = radicalBig (1 + 1 19 ) 2 18 38 + ( - 1 + 1 19 ) 2 20 38 0 . 998 (c) Let X be 2 if it lands on 1-12, - 1 otherwise. E ( X ) = 2( 12 38 )+( - 1) 26 38 = - 1 19 , SE ( X ) = radicalBig (2 + 1 19 ) 2 12 38 + ( - 1 + 1 19 ) 2 26 38 1 . 39 (d) Let X be 6 if it lands on 0,00,1,2, or 3, - 1 otherwise. E ( X ) = 6( 5 38 ) + ( - 1) 33 38 = - 3 38 , SE ( X ) = radicalBig (6 + 1 19 ) 2 5 38 + ( - 1 + 1 19 ) 2 33 38 2 . 37 2. For each of the following six-sided die, find the expected value and standard error of rolling the die once.
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