prob.part22.45_46 - 1.2. DISCRETE PROBABILITY DISTRIBUTIONS...

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Unformatted text preview: 1.2. DISCRETE PROBABILITY DISTRIBUTIONS 37 less than 85 percent lost one leg. What is the minimal possible percentage of those who simultaneously lost one ear, one eye, one hand, and one leg? 22 *17 Assume that the probability of a “success” on a single experiment with n outcomes is 1 /n . Let m be the number of experiments necessary to make it a favorable bet that at least one success will occur (see Exercise 1.1.5). (a) Show that the probability that, in m trials, there are no successes is (1- 1 /n ) m . (b) (de Moivre) Show that if m = n log2 then lim n →∞ 1- 1 n m = 1 2 . Hint : lim n →∞ 1- 1 n n = e- 1 . Hence for large n we should choose m to be about n log 2. (c) Would DeMoivre have been led to the correct answer for de M´ er´ e’s two bets if he had used his approximation? 18 (a) For events A 1 , . . . , A n , prove that P ( A 1 ∪ ···∪ A n ) ≤ P ( A 1 ) + · · · + P ( A n ) ....
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prob.part22.45_46 - 1.2. DISCRETE PROBABILITY DISTRIBUTIONS...

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