Question

Suppose the four engines of a commercial aircraft are arranged to operate independently and that the probability of in-flight failure of a single engine is .01. What is the probability of the following events on a given flight? a. No failures are observed. b. No more than one failure is observed.

Short answer

Answer: The probability of no failures during the flight is approximately 0.9606, and the probability of no more than one failure is approximately 0.9994.

Step-by-step solution

Calculate the probability of no failures

To find the probability of observing no failures, use the binomial probability formula with \(n=4\), \(x=0\), and \(p=0.01\): \(P(0) = \binom{4}{0}(0.01)^0(1-0.01)^{4-0}\) \(P(0) = \binom{4}{0}(1)(0.99)^{4}\) \(P(0) = 1(0.99)^{4}\) \(P(0) = 0.96059601\) So, the probability of no failures occurring during the given flight is approximately 0.9606.

Calculate the probability of exactly one failure

To find the probability of observing exactly one failure, use the binomial probability formula with \(n=4\), \(x=1\), and \(p=0.01\): \(P(1) = \binom{4}{1}(0.01)^1(1-0.01)^{4-1}\) \(P(1) = \binom{4}{1}(0.01)(0.99)^{3}\) \(P(1) = 4(0.01)(0.970299)\) \(P(1) = 0.03880852\) So, the probability of exactly one failure during the given flight is approximately 0.0388.

Calculate the probability of no more than one failure

To find the probability of observing no more than one failure, simply add the probabilities for no failures and exactly one failure: \(P(0 \text{ or } 1) = P(0) + P(1)\) \(P(0 \text{ or } 1) = 0.96059601 + 0.03880852\) \(P(0 \text{ or } 1) = 0.99940453\) So, the probability of observing no more than one failure during the given flight is approximately 0.9994.

Key concepts

Independent events

In probability, the concept of independent events is foundational. Two events are considered independent if the occurrence of one does not affect the occurrence of the other. When it comes to problems involving independent events, like the one with the aircraft engines, this means that the failure of one engine does not depend on whether any of the other engines have failed.

For example, if we consider each engine failure as an individual event, the probability of each failure remains constant for each engine irrespective of the others. This independence simplifies calculations, because the joint probability of multiple independent events is simply the product of their individual probabilities.

Understanding independence helps simplify complex real-world problems, making it easier to calculate overall probabilities in scenarios where many events could potentially occur.

Probability of failure

The probability of failure in an event is an essential factor in analyzing risk, especially in technological and industrial settings like aviation. It quantifies the likelihood that a particular component or system will fail within a defined period.

In the given exercise, the probability of failure for each individual engine is stated as 0.01, meaning there is a 1% chance of any one engine failing during a flight. This measurement is crucial as it informs the overall safety analysis of the aircraft's flight operation.

When evaluating multiple items or systems, understanding each individual's probability of failure is key to anticipating potential adverse outcomes. It allows risk managers and engineers to set thresholds and design failsafes for scenarios where multiple components might fail.

Binomial distribution

The binomial distribution is a statistical method used to model the number of successful outcomes in a set of independent trials, where each trial has the same probability of success. It's particularly handy for calculating probabilities like those seen in the aircraft engine exercise.

With four engines that can each independently fail or succeed, the binomial distribution comes into play by helping determine the total likelihood of different numbers of failures—ranging from zero to four. The formula for the binomial probability is: \( P(k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the number of trials (engines), \( k \) is the number of successful outcomes (failures in this case), and \( p \) is the probability of success (or a single engine failing).

This mathematical model is extremely valuable because it accounts for all possible outcomes, providing a way to calculate cumulative probabilities, such as the likelihood of no more than one failure. Such calculations are indispensable in assessing risks and preparing contingency plans.