Unformatted text preview: P n,m = n-m 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 n-1 ,m + m n + m P n,m-1 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 = n-m 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 n-1-m n-1 + m + m n + m n-m + 1 n + m-1 = n-m 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...
View Full Document
- Fall '07
- Natural number, Theoretical Exercise