The [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution) has a mode of $0$, which, [according to Wikipedia](https://en.wikipedia.org/wiki/Mode_%28statistics%29), means that $0$ "is the value that is most likely to be sampled". This is not what I would expect, given that the exponential distribution "describes the time between events in a Poisson process".

So, say me getting phone calls is a Poisson process and I get a call every hour on average. How is it that $0$ is the most likely sample? My intuition tells me that $0$ is one of the most *unlikely* samples, or at least much less likely than some number around $1$ hour.