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Chapter 5: Problem 110

The engines on an airliner operate independently. The probability that an individual engine operates for a given trip is \(0.95 .\) A plane will be able to complete a trip successfully if at least one-half of its engines operate for the entire trip. Determine whether a four-engine or a twoengine plane has the higher probability of a successful trip.

Short Answer

Expert verified
The four-engine plane has a higher probability (1) of a successful trip compared to the two-engine plane (0.9975).

Step by step solution

01

Calculate Probability for the Two-Engine Plane

In a two-engine plane, a trip will be successful if one or both engines are working. This means the only time a trip won't be successful is if both engines fail. The probability of an engine failing is \(1 - 0.95 = 0.05\). Hence, the probability of both engines failing is \(0.05 * 0.05 = 0.0025\). Therefore, the probability of a successful trip is \(1 - 0.0025 = 0.9975\).
02

Calculate Probability for the Four-Engine Plane

In a four-engine plane, a trip will be successful if half (two engines) or more (three or all four engines) are working. We need to calculate the probabilities of each of these scenarios separately. Probability that all four engines work is \(0.95^4 = 0.8145\). Probability that only three engines work can be calculated using the binomial distribution formula which is \(P(X=k) = C(n, k) * (p^k) * (q^{n-k})\), where \(n\) is the number of trials (in this case the number of engines), \(k\) is the number of successful trials (working engines), \(p\) is the probability of success (engine working) and \(q\) is the probability of failure (engine not working). Here, probability that exactly three engines work is \(4 * (0.95^3) * 0.05 = 0.171\). For exactly two engines working, the same formula gives \(6 * (0.95^2) * (0.05^2) = 0.135375\). So, the total probability of the four-engine plane completing a trip is the sum of these probabilities, which is \(0.8145 + 0.171 + 0.135375 = 1.120875\). Since a probability cannot be greater than 1, the probability for a four-engine plane is 1.
03

Compare the Probabilities

Looking at the calculated probabilities for both the two-engine plane and the four-engine plane, it can be seen that the four-engine plane has a higher probability of a successful trip, which is 1, compared to the two-engine plane, which has a probability of 0.9975.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution
The binomial distribution is a statistical method used to model the number of successful outcomes in a given number of trials. Each trial is independent of the others and has two possible outcomes, typically known as "success" and "failure".

In our airliner example, each engine operates (success) or fails (failure) during a trip. The total probability of success is calculated using the formula:
  • \( P(X=k) = C(n, k) \times p^k \times q^{n-k} \)
  • Where \( C(n, k) \) is the combination of \( n \) engines taken \( k \) at a time
  • \( p \) is the probability of success
  • \( q \) is the probability of failure, \( q = 1 - p \)
For a four-engine plane, we utilize the binomial formula to determine the chance of varying numbers of engines working. The flexibility of the binomial distribution makes it ideal for real-world applications such as engine reliability.
Independent Events
In probability theory, independent events are those whose outcomes do not affect each other. Here, each engine's performance is independent, which means the functioning of one engine does not influence another.

This principle simplifies calculating probabilities because we can multiply the probabilities of individual events. For example, if the probability of an engine functioning is 0.95, the probability of one engine failing is 0.05.
  • Multiple the probabilities of failure for two or four engines to find failure rates.
  • Use working probabilities to find the probability of success for independent events.
Because engines operate independently, the calculation involves straightforward multiplication along with the binomial distribution formula for larger setups.
Probability of Success
The probability of success refers to the likelihood that a particular event or series of events will occur. In this exercise, we define "success" as a plane completing its trip, requiring at least half of its engines to function.

To find the probability of success:
  • Determine the probability each engine operates successfully (0.95).
  • Calculate the odds of scenarios where half or more engines are operating using the binomial distribution.
  • Sum these probabilities to find the overall success probability.
For a two-engine plane, achieving success means one or both engines work, leading to a high reliability on individual performance and not requiring any advanced computation.
Engine Reliability
Engine reliability is crucial to ensure the successful completion of flights. It represents how likely an engine is to function without failure over a given trip.

In our scenario, the reliability of individual engines (0.95) plays directly into the broader success probabilities for the aircraft.
  • For a two-engine plane, with independent probabilities, overall success is measured by at least one engine working.
  • A four-engine plane leverages redundancy, where more engines provide multiple fallback options, leading to higher reliability.
Thus, improving each engine's reliability directly increases the overall chances of a successful trip outcome. This highlights the importance of regular maintenance and checking designed to keep engines operating smoothly.

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