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Chapter 4: Problem 150

A trimotor plane has three engines-a central engine and an engine on each wing. The plane will crash only if the central engine fails and at least one of the two wing engines fails. The probability of failure during any given flight is \(.005\) for the central engine and \(.008\) for each of the wing engines. Assuming that the three engines operate independently, what is the probability that the plane will crash during a flight?

Short Answer

Expert verified
The probability that the plane will crash during a flight is \(0.00007968\).

Step by step solution

01

Identify the probabilities

From the problem, we have that the probability of the central engine failing is \(0.005\). The probability of each of the wing engines failing is \(0.008\). Since the failure of each engine is considered an independent event, these probabilities can be directly used.
02

Calculate the probability of at least one wing engine failing

The probability of at least one wing engine failing is the probability of only the left engine failing, plus the probability of only the right engine failing, minus the probability of both engines failing. So, \(P(L or R) = P(L) + P(R) - P(L)*P(R) = 0.008 + 0.008 - 0.008*0.008 = 0.015936\). Note that we subtract the probability of both engines failing because we have counted it twice.
03

Calculate the probability of the plane crashing

The plane will crash if the central engine fails and at least one wing engine fails. So, the combined probability is the product of the probability of the central engine failing and the probability of at least one wing engine failing, i.e., \(P(Crash) = P(C) * P(L or R) = 0.005 * 0.015936 = 0.00007968\).

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