Let (An : n > 1) be a sequence of events in a r-algebra & on which is defined a probability Pr. Showthatfor each n E N and alsonPr(U=1 Ai) ≤ Σ Pr(A;)i=1Pr(U, Ai) ≤ Σ Pr(Ai).i=1

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Answered: Let (An : n > 1) be a sequence of…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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1. (Independence). (i) Recall that given events A, B,C, we have that P(AUBUC) = P(A)+ P(B)+ P(C) – P(An B) – P(AnC)– P(BnC) + P(AN BnC). (4.1) Prove that whenever events A, B,C are mutually independent, we have that P(AUBUC) = 1– P(A^)P(B^)P(C*) = 1– (1– P(A)) · (1 – P(B)) · (1 – P(C)) (and the probability in question can be calculated considerably faster than with the use of the formula (4.1)). (ii) The probability that A hits a target is 2/5, the probability that B hits it is 5/9, and the probability that C hits the target is 3/7. Use (i) to find the probability that the target will be hit if A, B, and C each shoot at the target. (iii) Suppose that the probability that a soldier firing his personal weapon hits an enemy warplane is p > 0. Show then, arguing as in (i), that the probability that the warplane is hit at least once when n > 2 soldiers shoot at it is 1- (1 — р)". (4.2) Evaluate the probability in (4.2) in the case when p = 0.0006 and n = 750, and then round your result to a…

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Let (An : n > 1) be a sequence of events in a r-algebra æ on which is defined a probability Pr. Show that for each n E N and also n Pr(V=1 Ai) = Σ Pr(Ai) i=1 Pr(Up, Ai) ≤ Σ Pr(A;). i=1