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Unformatted text preview: IEOR 172: Probability and Risk Analysis for Engineers, Fall 2007 Solution of Review Questions for Final 1. There are three coins in a box. One is a twoheaded coin; another is a fair coin; and the third is a biased coin that comes up heads 60% of the time. When one of the three coins is selected at random and flipped twice, it showed heads both times. What is the probability that the coin selected is the twoheaded coin? By Bayes’s rule P { twoheaded  HH } = P { HH  twoheaded } P { twoheaded } P { HH  twoheaded } P { twoheaded } + P { HH  fair } P { fair } + P { HH  biased } P { biased } = 1 · 1 3 1 · 1 3 + 1 4 · 1 3 + 6 10 · 6 10 · 1 3 = 100 161 . 2. Suppose we have two coins. When coin 1 is flipped, it lands heads with probability .4; when coin 2 is flipped, it lands heads with probability .6. One of these two coins is randomly chosen and flipped 10 times. (a) What is the probability that exactly 8 of the 10 flips land on heads? Let E denote the event that exactly 8 of the 10 flips land on heads P( E ) = P { E  coin 1 selected } P { coin 1 selected } + P { E  coin 2 selected } P { coin 2 selected } = 10 8 ! ( . 4) 8 ( . 6) 2 1 2 + 10 8 ! ( . 6) 8 ( . 4) 2 1 2 = . 0658 (b) Given that the first of these 10 flips lands on heads, what is the conditional probability that exactly 8 of the 10 flips land on heads? Let E denote the event that exactly 8 of the 10 flips land on heads, let F denote the event that the first of these 10 flips lands on heads. P { E  F } = P { EF } P { F } = P { EF  coin 1 selected } P { coin 1 selected } + P { EF  coin 2 selected } P { coin 2 selected } P { F  coin 1 selected } P { coin 1 selected } + P { F  coin 2 selected...
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