Question

The increased number of small commuter planes in major airports has heightened concern over air safety. An eastern airport has recorded a monthly average of five near misses on landings and takeoffs in the past 5 years. a. Find the probability that during a given month there are no near misses on landings and takeoffs at the airport. b. Find the probability that during a given month there are five near misses. c. Find the probability that there are at least five near misses during a particular month.

Short answer

b) What is the probability of having exactly five near misses during a particular month? c) What is the probability of having at least five near misses during a particular month?

Step-by-step solution

Identify the mean (λ) of the Poisson distribution

In this problem, we are given the monthly average of five near misses. This means that the mean (λ) for our Poisson distribution is 5.

Calculate the probability of no near misses (a)

To find the probability of no near misses during a given month, we will use the Poisson probability mass function for x = 0: P(X = 0) = e^{-λ} * (λ^0) / 0! where X is the number of near misses, λ is the mean, and e is the base of the natural logarithm (approximately 2.71828). Plug in the values: P(X = 0) = e^{-5} * (5^0) / 0! Calculate the probability: P(X = 0) ≈ 0.0067

Calculate the probability of exactly five near misses (b)

To find the probability of exactly five near misses during a given month, we will use the same Poisson probability mass function, but this time for x = 5: P(X = 5) = e^{-λ} * (λ^5) / 5! Plug in the values: P(X = 5) = e^{-5} * (5^5) / 5! Calculate the probability: P(X = 5) ≈ 0.1755

Calculate the probability of at least five near misses (c)

To find the probability of having at least five near misses during a particular month, we need to find the probability of having 5 or more near misses. We can do this by finding the cumulative probability of having 0 to 4 near misses and then subtracting it from 1. We'll use the Poisson probability mass function again: P(X ≥ 5) = 1 - P(X < 5) This means we need to calculate P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4), and then subtract that from 1. After plugging in the values and using the Poisson probability mass function for each value of x, we get: P(X ≥ 5) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)] ≈ 1 - [0.0067 + 0.0337 + 0.0842 + 0.1404 + 0.1755] ≈ 0.5595 So, the probability of at least five near misses during a particular month is approximately 0.5595.

Key concepts

Probability Mass Function

When dealing with discrete variables, like the number of near misses at an airport, we often rely on a probability mass function (PMF) to compute the probabilities of the possible outcomes. In the context of a Poisson distribution, the PMF allows us to calculate the likelihood of observing a specific number of events—such as near misses—within a fixed period, given that these events occur with a known average rate and independently of the time since the last event.

The generic formula for the Poisson PMF where the average rate is λ (lambda) and the number of events is x is:
\[ P(X = x) = \frac{e^{-\lambda} \cdot \lambda^x}{x!} \]
This formula reflects the probability that exactly x events will occur. To effectively use the PMF, we should identify the average rate (λ), determine the exact number of events (x), and use the PMF to compute the likelihood of these events.

Cumulative Probability

Cumulative probability is another vital concept in statistics, particularly useful when we want to know the probability of a variable falling within a certain range or below a specific value. For the Poisson distribution, cumulative probability sums the probabilities of all outcomes up to (and including) a particular point.

In practice, calculating cumulative probability in a Poisson distribution often involves summing individual probabilities obtained from the PMF for each outcome from 0 up to a maximum number of events. This calculation can become cumbersome if done manually for a large range of values, which is why statistical software or tables often assist in these computations.

If the exercise asks for the probability of at least five near misses, the cumulative probability approach would consider the complement, that is, 1 minus the sum of probabilities of having fewer than five near misses, providing a cumulative assessment for five or more occurrences.

Mean of Poisson Distribution

The mean of a Poisson distribution is a critical parameter denoted by λ, which is equal to the expected number of events occurring in a fixed interval of time or space. This mean value is not only the average but also the variance of the distribution, making it unique amongst probability distributions.

Knowing the mean is essential for determining the probabilities using the PMF. In the given exercise, the mean value is directly stated: five near misses per month. This single parameter can give us a clear understanding of what is considered normal within the given context and enables us to calculate probabilities for different scenarios—such as no near misses or exactly five near misses—using the PMF. It’s a prime example of how one statistic—mean—shapes the entire probability landscape for a Poisson-distributed random variable.