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

To summarize: a) The probability that no failures are observed among the four engines is approximately 0.9616. b) The probability that no more than one failure is observed among the four engines is approximately 0.9988.

Step-by-step solution

Calculate probability of no failures (k=0) #

We can use the binomial probability formula P(X=0) = C(4, 0) * 0.01^0 * (1-0.01)^(4-0): P(X=0) = C(4, 0) * 0.01^0 * 0.99^4 Since C(4, 0) = 1 and 0.01^0 = 1, we have: P(X=0) = 1 * 1 * 0.99^4 ≈ 0.9616 #a# The probability that no failures are observed is about 0.9616.

Calculate probability of exactly one failure (k=1) #

Now we can use the binomial probability formula for P(X=1) = C(4, 1) * 0.01^1 * (1-0.01)^(4-1): P(X=1) = C(4, 1) * 0.01^1 * 0.99^3 Since C(4, 1) = 4, we have: P(X=1) = 4 * 0.01 * 0.99^3 ≈ 0.0372 #b# The probability of no more than one failure means the probability of no failures (k=0) or one failure (k=1). We should add the probabilities calculated: P(no more than one failure) = P(X=0) + P(X=1) ≈ 0.9616 + 0.0372 ≈ 0.9988 The probability that no more than one failure is observed is about 0.9988.

Key concepts

Probability of Independent Events

Understanding the probability of independent events is crucial for evaluating scenarios where multiple outcomes do not influence each other. In terms of probability, events are independent if the occurrence of one does not affect the probability of the other. A common example is flipping a coin; whether you get heads or tails on the first flip does not impact the result of the second flip.

If we have two independent events, A and B, the probability of both occurring (denoted as P(A and B)) can be calculated by multiplying the probabilities of each event occurring separately:
\[ P(A \text{ and } B) = P(A) \times P(B) \].

In our aircraft engine failure problem, the engines operate independently. Therefore, the failure or success of one engine does not impact the others. Consequently, this property allows us to use the multiplication rule to find the probability of multiple events happening at the same time (e.g., more than one engine failing during a flight).

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take on one of two independent outcomes across a number of trials or events. It's particularly useful for modeling situations where there are a fixed number of identical trials, each with the same probability of success, and each trial (also called a Bernoulli trial) is independent of the others.

The probability of getting exactly k successes in n trials is given by the formula:
\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \],where:\( p \) is the probability of success on a single trial, \( \binom{n}{k} \) is a binomial coefficient computed as \( \frac{n!}{k!(n-k)!} \), and \( X \) is the random variable representing the number of successes.

For the aircraft engine example, the probability of any single engine failing is 0.01, and the binomial distribution helps calculate the likelihood of various failure scenarios over the four independent trials represented by the engines.

Calculating Probabilities

Calculating probabilities involves identifying the likelihood of various outcomes. When applying the binomial distribution, it's vital to first understand the specific question being asked. For instance, in the provided example, we are interested in two scenarios: no engine failures and no more than one engine failure.

To calculate these probabilities, we use the binomial formula as shown in the step-by-step solution. You can also use technology like graphing calculators or statistics software for more complex problems. Remember, you'll need to know or calculate the binomial coefficients, which tell us how many ways k successes can occur within n trials, and then apply the probabilities of success and failure accordingly.

If a question asks for the probability of 'at most' or 'no more than' a certain number of successes, as in the second part of the engine failure example, you'll add the probabilities of all outcomes up to that number. In this case, you combine the probability of no failures and the probability of exactly one failure to find the total likelihood of one or fewer failures in flight.