CS 2051: Honors Discrete Math

Homework 10Due Tuesday, November 26, 2013

1. A complex system with 100,000 different components contains 10 critical ones, each failingwith probability 1/1000 and 100 secondary ones, each failing with probability 1/100. It isknown that secondary failures are pairwise independent. The system crashes if one ciritcalcomponent fails or two secondary components fail.

(a) Under the information given, determine the best upper and lower bounds on the prob-ability that the system crashes.

(b) In addition to the above provided information, assume that failure of critical componentsis independent of failure of secondary components, and repeat part (a).

(c) Further, in addition assume that failure of critical components is mutually independent,and repeat part (a).

2. Two numbers x and y are selected at random from the set {1, . . . , 100}. Let A,B,C be thefollowing events:

A = {x > 50}, B = {y > 50}, C = {x > y}.

Which pairs of events are independent and which not?

3. If you repeatedly flip a fair coin, what is the probability that you see the sequence HTT beforeyou see the sequence HHT?

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