Unformatted text preview: P n,m = nm n + m ,n ≥ m Theoretical Exercise 3.21 d. P n,m = P (A always ahead) = P (A always ahead  A receives last vote) n n + m + P (A always ahead  B receives last vote) m n + m = n n + m P n1 ,m + m n + m P n,m1 Theoretical Exercise 3.21 e. The conjecture of (c) is true when n + m = 1 ( n = 1 ,m = 0). Assume by induction that P n,m = nm n + m when n + m = k . Now suppose that n + m = k + 1. By (d) and the induction hypothesis, we have that P n,m = n n + m n1m n1 + m + m n + m nm + 1 n + m1 = nm n + m (1) which completes the proof. Theoretical Exercise 3.22 If the weather is dry on January 1st, then P = P (dry on Jan 1st) = 1 and, conditioning 3...
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 Fall '07
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 Natural number, Theoretical Exercise

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