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MIT6_042JS10_lec35_prob

# MIT6_042JS10_lec35_prob - 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, 118 minutes 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.

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