Final answer:
The number of airplanes that arrive and depart the airport per hour is approximately 104. The probability that there are exactly 100 arrivals and departures in one hour is approximately 0.0671. The probability that there are at most 100 arrivals and departures in one hour is approximately 0.9798.
Step-by-step explanation:
a. To find the number of airplanes that arrive and depart the airport per hour, we need to know the average number of arrivals and departures per day and divide it by 24 (since there are 24 hours in a day). The information provided states that there are 2,500 arrivals and departures each day, so the number of airplanes that arrive and depart the airport per hour is:
2,500 / 24 = 104.1666667
Therefore, approximately 104 airplanes arrive and depart the airport per hour.
b. To find the probability that there are exactly 100 arrivals and departures in one hour, we need to use the Poisson distribution. The formula for the Poisson distribution is:
P(x;λ) = (e^(-λ) * λ^x) / x!
Where x is the number of arrivals and departures (100 in this case) and λ is the average number of arrivals and departures per hour (104.1666667). Plugging in the values:
P(100; 104.1666667) = (e^(-104.1666667) * 104.1666667^100) / 100!
Using a calculator or software, we can find that the probability is approximately 0.0671.
c. To find the probability that there are at most 100 arrivals and departures in one hour, we need to sum up the probabilities of having 0, 1, 2, ..., 100 arrivals and departures. This can be calculated using the Poisson distribution as well. Plugging in the values into the formula for each number, we can add them up to get the probability:
P(0; 104.1666667) + P(1; 104.1666667) + P(2; 104.1666667) + ... + P(100; 104.1666667)
Using a calculator or software, we can find that the probability is approximately 0.9798.