Answer:
0.0498 = 4.98%
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
![P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20%5Cfrac%7Be%5E%7B-%5Cmu%7D%2A%5Cmu%5E%7Bx%7D%7D%7B%28x%29%21%7D)
In which
x is the number of sucesses
e = 2.71828 is the Euler number
is the mean in the given time interval.
Inquiries arrive at a record message device according to a Poisson process of rate 15 inquiries per minute.
Each minute has 60 seconds.
So a rate of 1 inquire each 4 seconds.
The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately
Mean of 1 inquire each 4 seconds, so for 12 seconds ![\mu = \frac{12}{4} = 3](https://tex.z-dn.net/?f=%5Cmu%20%3D%20%5Cfrac%7B12%7D%7B4%7D%20%3D%203)
This probability is P(X = 0).
![P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20%5Cfrac%7Be%5E%7B-%5Cmu%7D%2A%5Cmu%5E%7Bx%7D%7D%7B%28x%29%21%7D)
![P(X = 0) = \frac{e^{-3}*3^{0}}{(0)!} = 0.0498](https://tex.z-dn.net/?f=P%28X%20%3D%200%29%20%3D%20%5Cfrac%7Be%5E%7B-3%7D%2A3%5E%7B0%7D%7D%7B%280%29%21%7D%20%3D%200.0498)