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Discrete Mathematics with Graph Theory (3rd Edition) 198

Discrete Mathematics with Graph Theory (3rd Edition) 198 -...

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196 Solutions to Exercises 20. [BB] We use induction on n, the strong form. For n = 1, the result is clear. If n = 2, P(A I U A2) = P(A I) + P(A 2) - P(A I n A2) ~ P(A I) + P(A2)' Now assume n > 2 and the result holds for all integers k with 1 ~ k < n. We have P(A I U A2 U ... U An) = P((A I U A2 U An-I) U An) ~ P(A I U A2 U ... U An-I) + P(An) ~ (P(A I) + P(A 2) + ... + P(A n- I) + P(An) By the Principle of Mathematical Induction, the result follows. using the k = 2 case using the k = n - 1 case. 21. We use induction on n, the strong form. For n = 1, the result is clear. If n = 2, P(A I n A2) = P(A I) + P(A 2) - P(A I U A2) ~ P(A I) + P(A 2) - 1. Now assume n > 2 and the result holds for all integers k with 1 ~ k < n. We have P(A I n A2 n ... nAn) = P((A I n A2 n An-I} nAn) ~ P(A I n A2 n ... nAn-I) + P(An) - 1 ~ (P(A I) + P(A 2) + ... + P(A n- I) - (n - 2) + P(An) - 1 = P(A I) + P(A 2) + ... + P(An) - (n - 1). By the Principle of Mathematical Induction, the result follows. 22. (a) 365(364)(363)··· (365 - (n - 1)) (365)n using the k = 2 case using the k = n - 1 case (b) When n = 23, the fraction in (a) is .492703 and this value is closer to .5 than with any other n. 23. nPx is the probability of the event "a person aged x survives to age x + n" while mPx+n is the probability of the event
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