midterm2sol

# You do not need to simplify the expression just write

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Unformatted text preview: obability that a player uses CS 70, Fall 2013, Midterm #2 2 Name: SID: steroids if he tested positive? You do not need to simplify the expression - just write it in terms of p and q. The probability is pq pq+(1− p)(1−q) . Solution: Let A be the event that an NCAA football player uses steroids. Let B be the event that the test result is positive. Then we are given: Pr[A] = q Pr[B|A] = p ¯ Pr[B|A] = 1 − p We would like to ﬁnd Pr[A|B]. By the total probability rule: ¯ ¯ Pr[B] = Pr[B|A] Pr[A] + Pr[B|A] Pr[A] = pq + (1 − p)(1 − q). And by Bayes’ rule: Pr[A|B] = CS 70, Fall 2013, Midterm #2 Pr[B|A] Pr[A] pq = . Pr[B] pq + (1 − p)(1 − q) 3 Name: SID: 2. (15 points) Even or odd. Suppose you ﬂip a biased coin with P[H ] = p. Let En be the event you get 2n an even number of H’s in n tosses of the coin. Prove by induction on n that P[En ] = 1 + (1−2 p) for 2 n ≥ 1. Solution: Base Case: n = 1. E1 is the event that you obtain tails on the ﬁrst day. Pr[E1 ] = 1 − p = 1 + 1−2 p as 2 2 claimed. n−1 Inductive Hypothesis: Assume that Pr[En−1 ] = 1 + (1−22p) 2 . n 2 Inductive Step: Prove that Pr[En ] = 1 + (1−2 p) . 2 We can use the total probability rule: ¯ ¯ Pr[En ] = Pr[En |En−1 ] Pr[En−1 ] + Pr[En |En−1 ] Pr[En−1 ]. Observe that Pr[En |En−1 ] is...
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## This document was uploaded on 01/28/2014.

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