Poisson process. Time between two events. Suppose that people immigrate to a territory according to a Poisson process with a $\lambda =$ rate of 1 per day. What is the probability that the time between the tenth and eleventh exceeds two days?

There are two approaches, through the Poisson and through the associated exponential.

The number of events in two days is a Poisson random variable $Y$ with parameter $2\lambda$. The probability that $Y=0$ is $e^{-2\lambda}$.

Or else the waiting time $W$ is an exponentially distributed random variable with parameter $2$. The probability that $W\gt 2$ is $$\int_2^\infty \lambda e^{-\lambda t}\,dt.$$ Integrate (or remember the cdf of the exponential). We get $e^{-2\lambda}$.