An average of 1.4 private airplanes arrive per hour at an airport.
Let x denote the number of private airplanes that will arrive at this airport during a given hour. Calculate the probability distribution of x.
Find the probability that x = 4.
Using a Poisson distribution:
P(x) = (e^-μ) (μ^x) / x!
e = 2.71828
µ = 1.4
x = 4
Substitute and calculate.
To calculate the probability distribution of x, we need to use the Poisson probability formula, which is given by:
P(x; μ) = (e^(-μ) * μ^x) / x!
where P(x; μ) is the probability function of x, μ is the average number of arrivals per hour, e is Euler's number approximately equal to 2.71828, x is the number of arrivals, and x! represents the factorial of x.
In this case, the average number of private airplanes that arrive per hour is 1.4, so μ = 1.4.
To find the probability that x = 4, we substitute these values into the formula:
P(4; 1.4) = (e^(-1.4) * 1.4^4) / 4!
Calculating this equation will give us the desired probability.
To calculate the probability distribution of x, we can use the Poisson distribution since it is appropriate for modeling rare events. The formula for the Poisson distribution is:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
- P(x; λ) is the probability of x events occurring
- λ is the average number of events occurring in a given time period
In this case, the average number of private airplanes arriving per hour is 1.4 (λ = 1.4).
To find the probability that x = 4, we can substitute x = 4 and λ = 1.4 into the Poisson distribution formula:
P(4; 1.4) = (e^(-1.4) * 1.4^4) / 4!
Now let's calculate this probability step by step:
1. Calculate e^(-1.4):
e^(-1.4) ≈ 0.2466
2. Calculate 1.4^4:
1.4^4 ≈ 3.8416
3. Calculate 4! (4 factorial):
4! = 4 * 3 * 2 * 1 = 24
4. Substitute the values into the formula and calculate the probability:
P(4; 1.4) = (0.2466 * 3.8416) / 24 ≈ 0.0393
Therefore, the probability that x = 4 is approximately 0.0393 or 3.93%.
Note: The Poisson distribution assumes that the events occur independently and at a constant rate.