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Examples of calculations of Probability and Reliability

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Q1. A smoke detector is routinely inspected. 80% of the detectors found inoperative had experienced a power surge, and 10% of those found in operating condition had experienced a power surge. 20% of the detectors inspected have failed. What is probability of a detector failing given it experiences a power surge?

Q2. The time to failure of a typical household refrigerator has the following PDF:
f(t)=t/18 0≤t≤6 yr
What is the reliability of the refrigerator for the first year?
What is the probability that a refrigerator will fail during the first month?
Given the refrigerator has survived its first month, what is the probability that it will survive the rest of the year?

Q3. Wadih Company manufactures gizmos for use on widgets. The time to failure, in years, of these gizmos has the following PDF:
f(t)= 200/〖(t+10)〗^3 for t≥0
Derive the reliability function and determine the reliability for the first year of operation.
Compute the MTTF.
What is the design life for a reliability of 0.95?
Median time of failure

Q4. The time to failure, in years, of a product has the following PDF:

f(x)={█(a(x^2+1), 0<x<10@0, otherwise)┤
Find value of a.

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