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MIT6_042JS10_lec35_sol

MIT6_042JS10_lec35_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science May 3 Prof. Albert R. Meyer revised May 3, 2010, 616 minutes Solutions to In-Class Problems Week 13, Mon. Problem 1. Suppose there is a system with n components, and we know from past experience that any partic- ular component will fail in a given year with probability p . That is, letting F i be the event that the i th component fails within one year, we have Pr { F i } = p for 1 i n . The system will fail if any one of its components fails. What can we say about the probability that the system will fail within one year? Let F be the event that the system fails within one year. Without any additional assumptions, we can’t get an exact answer for Pr { F } . However, we can give useful upper and lower bounds, namely, p Pr { F } ≤ np. (1) We may as well assume p < 1 /n , since the upper bound is trivial otherwise. For example, if n = 100 and p = 10 5 , we conclude that there is at most one chance in 1000 of system failure within a year and at least one chance in 100,000. Let’s model this situation with the sample space S ::= P ( { 1 , . . . , n } ) whose outcomes are subsets of positive integers n , where s ∈ S corresponds to the indices of exactly those components that fail within one year. For example, { 2 , 5 } is the outcome that the second and fifth components failed within a year and none of the other components failed. So the outcome that the system did not fail corresponds to the emptyset, . (a) Show that the probability that the system fails could be as small as p by describing appropriate probabilities for the outcomes. Make sure to verify that the sum of your outcome probabilities is 1. Solution. There could be a probability p of system failure if all the individual failures occur to- gether. That is, let Pr {{ 1 , . . . , n }} ::= p , Pr {∅} ::=1 p , and let the probability of all other outcomes be zero. So F i = { s ∈ S | i s } and Pr { F i } = 0+0+ · · · +0+ Pr {{ 1 , . . . , n }} = Pr {{ 1 , . . . , n }} = p .
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