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midterm_with_solutions

midterm_with_solutions - Basic Probability Summer 2011 NYU...

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Basic Probability Summer 2011 NYU Courant Institute Midterm Exam with Solutions 1. Suppose that an airplane engine will fail, when in flight, with probability 1 - p indepen- dently from engine to engine; suppose that the airplane will make a successful flight if at least 50% of its engines remain operational. If p = 3 / 4, which is preferable, a four-engine plane or a two-engine plane? What about if p = 1 / 2? Let X be the binomial variable which is the number of engines that don’t fail. For the first part we have p = 3 / 4 and n = 4 and we want P ( X 2) = P ( X = 2) + P ( X = 3) + P ( X = 4) = 4 2 (3 / 4) 2 (1 / 4) 2 + 4 3 (3 / 4) 3 (1 / 4) 1 + 4 4 (3 / 4) 4 = . 9492 . For the two engine plane we have n = 2 and we want P ( X 1) = P ( X = 1) + P ( X = 2) = 2 1 (3 / 4) 1 (1 / 4) 1 + 2 2 (3 / 4) 2 = . 9375 So the four-engine plane is more likely to have a successful flight. Now, what if p = 1 / 2? For n = 4 we get P ( X 2) = P ( X = 2) + P ( X = 3) + P ( X = 4) = 4 2 (1 / 2) 2 (1 / 2) 2 + 4 3 (1 / 2) 3 (1 / 2) 1 + 4 4 (1 / 2) 4 = . 6875 and for n = 2 we get P ( X 1) = P ( X = 1) + P ( X = 2) = 2 1 (1 / 2) 1 (1 / 2) 1 + 2 2 (1 / 2) 2 = . 75 so the two-engine plane is safer. The conclusion is that, depending on
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