Answer Happy

Problem 2 (25 Points) A sensor network monitors the operation of an industrial power plant. The alarm system is triggered when any of the monitored variables exceeds a certain threshold and enables the implementation of accident prevention and mitigation measures. Suppose this alarm system is designed using redundancy, where three sensors are placed in parallel as described in the diagram below. The alarm is triggered if at least 2 out of the three sensors detect that a threshold has been exceeded. Sensor 1 Sensor 2 Sensor 3 Suppose the sensors are identical and have a probability of failure of 0.1. We define a system success as P(S): "The alarm system successfully detects a hazard". The system described above is called a "k-out-of-n" in which 2-out-of-3 elements must work to achieve a system-level success. For this alarm system: a. Draw the corresponding Venn Diagram for P(S). (4 pts) b. Find P(S) using set theory. Calculate the probability of P(S). (8 pts) c. Calculate the probability of P(S) using a binomial distribution. (8 pts) d. Suppose that the failure of sensors 1 and 2 is dependent with P(S₂1S₁) = 0.1. What does this mean for the design's safety? What would happen if P (5₂|5₁) > 0.1? (Hint: Express P(S) using set theory.) (5 pts)