airline passengers arrive randomly and independently at the passenger screening facility at an airport. The mean arrival rate is 10 passengers per minute. What is the probability of no arrivals in a 15 second period.
The number of arrivals in a time interval of length $\displaystyle \tau$ has a Poisson distribution with expected number $\displaystyle \tau \times \rho$ where $\displaystyle \rho$ is the expected number per unit time.
Here the expected numbe per unit time is $\displaystyle 1/6$ of an arrival per second, so the expected number in $\displaystyle 15$ seconds is $\displaystyle 5/2$.
Hence the probability of $\displaystyle k$ arrivals in a $\displaystyle 15$ second interval is:
$\displaystyle
f(k,2.5) = \frac{2.5^k e^{-2.5}}{k!}
$
so if $\displaystyle k=0$, this is:
$\displaystyle
f(0,2.5) = \frac{2.5^0 e^{-2.5}}{0!}=e^{-2.5}\approx 0.082
$
RonL