Independent Events Example
Reliability of Systems (Conceptual)
Consider a structural system with two components in series. For the system to function, both components must work. If the failure of one component increases the load on the other, the events are dependent. If they fail independently (e.g., due to random material defects), they are independent.
Solved Problems
Problem 1: Bayes' Theorem in Non-Destructive Testing (Intermediate)
A structural engineer is inspecting welds.
- 2% of all welds are defective (). .
- A non-destructive test (NDT) is used to detect defects.
- The test correctly identifies a defective weld 95% of the time ().
- The test incorrectly identifies a good weld () as defective 10% of the time (False Positive, ).
If a weld tests positive (), what is the probability it is actually defective ()?
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Problem 2: Conditional Probability in Project Management (Basic)
A construction manager is evaluating the likelihood of delays. Historically, 40% of projects experience material supply delays (). If a project experiences a material supply delay, there is an 80% chance that the project will miss its deadline (). What is the probability that a project experiences a material supply delay AND misses its deadline?
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Problem 3: Independent Events in Redundant Systems (Advanced)
A critical pumping station has a main pump and a backup pump. The probability that the main pump fails during a storm is 0.05. If the main pump fails, the backup pump is automatically activated. The probability that the backup pump fails when activated is 0.10. Assume the failure mechanisms are independent. What is the probability that the station completely fails (both pumps fail)? If a third backup pump with a 0.15 failure rate is added, what is the new probability of complete failure?
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