Conditional Probability and Independence: Uses 2

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Page 2 - Calculating Conditional Probabilities: Independent Events

Example - Finding Probabilities for Equipment Failure

electrical plug Pacific Gas and Electric (PGE) provides the electricity for San Francisco. When construction crews repair a circuit, there is a check list of steps to follow before reconnecting the electrical power. Crews rarely make a mistake. However, to prevent such an error from bringing down the entire electrical grid, PGE installed circuit breakers to contain any resulting power outage to a small area. The probability these breakers fail is very small.

Assuming the crew's actions and the performance of the circuit breakers are independent events, these combine to make the probability of system failure quite small.

In December 1998, two crews made mistakes during the same period. A construction crew failed to remove the ground before reconnecting a circuit. Elsewhere another crew had failed to activate necessary circuit breakers since their installation. These two unrelated human errors combined to cause a power outage in the entire county of San Francisco lasting most of the day and costing over $400 million.

Engineers often include independent back-up systems to increase reliability and prevent equipment failure. If the circuit breakers had been activated would the probability of failure have been small?

Suppose the P(crew mistake) = 0.01 and P(circuit breaker fails) = 0.001 and these are independent events:
P(crew mistake and circuit breaker fails) = P(crew mistake)P(circuit breaker fails)
P(crew mistake and circuit breaker fails) = (0.01)P(0.001) = 0.00001 or 1/100,000.

Because the circuit breakers were deactivated, P(circuit breaker fails) = 1.
P(crew mistake and circuit breaker fails) = 0.01(1) = 0.01 = 1/100.

Note: Human error is difficult to predict when systems are designed:
Both the nuclear accidents at Three Mile Island and Chernobyl were caused by human error:

At Chernobyl, engineers disabled the safety system to run a test.

At Three Mile Island, after a $3 valve failed, the operators did not follow the established procedures (checklist) and made a series of errors, which nearly caused a disaster.

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