winter 2006 s230_test2solutions

winter 2006 s230_test2solutions - STAT 230 Name: Test #2...

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STAT 230 Test #2 Feb 1, 2006 3:30 – 4:15 pm Name: S I.D: UWUserid: Mark /31 Circle your instructor and time slot: 1. McLeish (10:30) 2. Zhu (12:30) 3. Drekic (1:30) [15] 1. A not very skillful dart player throws a dart at a dart board. The probability he hits the bull’s-eye is 0.10. Find: (a) the probability he gets his ±rst bull’s-eye on his seventh throw. What assumptions are you making? Assumptions : All throws are independent of one another and the same probability of hitting a bull’s-eye (i.e., p = 0 . 10) applies to each throw. P (he gets his ±rst bull’s-eye on his seventh throw) = (0 . 90) 6 (0 . 10) 0 . 053 (b) the probability he gets his second bull’s-eye on his tenth throw. Let X the number of failed attempts before getting his second bull’s-eye, so that X has a negative binomial distribution with k = 2 and p = 0 . 10. We want P ( X = 8) = ( 8+2 - 1 8 ) (0 . 10) 2 (0 . 90) 8 = ( 9 8 ) (0 . 10) 2 (0 . 90) 8 0 . 039 (c) the probability he gets at least two bull’s-eyes in ten throws. Let Y the number of bull’s-eyes obtained in ten throws, so that Y has a binomial distribution with n = 10 and p = 0 . 10. We want: P ( Y 2) = 10 X y =2 µ 10 y (0 . 10) y (0 . 90) 10 - y = 1 - 1 X y =0 µ 10 y (0 . 10) y (0 . 90) 10 - y 0 . 264
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[10] 2. A box contains 5 red cards and 5 green cards. First, a fair coin is tossed. If the result is
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winter 2006 s230_test2solutions - STAT 230 Name: Test #2...

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