Question 6) Consider the following event tree for an automobile system, where th

Question

Question 6) Consider the following event tree for an automobile system, where the car battery has failed. The dead battery is the internal event begins the scenario analysis. Find mean time between mission failure Pivotal Events Initiating Event Cables Connected Properly Jumper Cables Donor Battery Donor Battery Starts Car Outcomes Available Available YES (P 0.9) Car is jump started mission success YES (P 0.8) YES (P 0.7) NO (P 0.1) Car not started, mission failure YES (P-0.6) NO (P 0.2) Car nt started, possible damage, mission failure Dead Batte NO (P 0.3) Car not started, mission failure NO (?-04) Car not started, mission failure

Explanation / Answer

In this tree, from our perspective, there are only two outcomes: Mission failure and Mission success. We have to find the probability of mission failure to get the mean time between failure. For this we need to find the probability of each of the leaves of the tree. There are 5 probabilities we can find:

1. P(Car is jump started, mission success)

2. P(Car not started, mission failure)

3. P(Car not started, possible damage, mission failure)

4. P(Car not started, mission failure)

5. P(Car not started, mission failure)

But we are interested in only those leaves which correspond to mission failure. So let us find the leaves corresponding to 'mission failure' one by one starting from the bottom most. There are four of them. We note that all the intermediate events are independent.

1. P(Car not started, mission failure) = P(Dead battery) x P(No Jumper cables available)

                                                     = 0.1 x 0.4

                                                     = 0.04

2. P(Car not started, mission failure) = P(Dead Battery) x P(Jumper cables available) x P(No donor battery available)

                                                      = 0.1 x 0.6 x 0.3

                                                      = 0.018

3. P(Car not started, Possible damage, mission failure) = P(Dead Battery) x P(Jumper cables available) x P(Donor battery available) x P(Cables not connected properly)

                                                      = 0.1 x 0.6 x 0.7 x 0.2

                                                      = 0.0084

4. P(Car not started, Possible damage, mission failure) = P(Dead Battery) x P(Jumper cables available) x P(Donor battery available) x P(Cables connected properly) x P(Donor battery does not start car)

                                                      = 0.1 x 0.6 x 0.7 x 0.8 x 0.1

                                                      = 0.00336

Now, to find the probability of mission failure, we add these four probabilities.

P(Mission Failure) = 0.04 + 0.018 + 0.0084 + 0.00336

                           = 0.06976

Mean time between misison failure = 1/P(Mission Failure)

                                                   = 1/0.06976

                                                   = 14.3349

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