Final answer:
The student's question about the probability of observing at least 6 birds on an airport runway during peak hours with no delays is a problem that can be solved using the Poisson distribution. To find the probability of at least 6 birds, the cumulative probability of observing fewer than 6 is calculated and then subtracted from 1.
Step-by-step explanation:
The question asks about the probability of observing at least 6 birds on an airport runway during peak hours, assuming there are no flight delays. This situation can be modeled by a Poisson distribution, where the average number of birds observed in a given time period (e.g., during one hour) is known. The average rate (λ) is given as 6 birds per hour.
To find the probability of observing at least 6 birds, we can use the Poisson distribution formula:
The probability mass function (PMF) of a Poisson distribution for k events is given by:
P(X = k) = (λ^k * e^-λ) / k!
However, to find the probability of observing at least 6 birds (k ≥ 6), we need to calculate the sum of probabilities of observing 6, 7, 8, ... and so on, up to infinity. A more practical approach is to calculate the cumulative distribution up to 5 and subtract it from 1:
P(X ≥ 6) = 1 - P(X < 6)
P(X < 6) is the cumulative probability of observing fewer than 6 birds, which can be calculated by summing the probabilities from k = 0 to k = 5 using the PMF of the Poisson distribution. Therefore, the calculation would involve summing the PMFs for k = 0 to k = 5 and subtracting this sum from 1.