Given a Poisson process (e.g. radioactive decay) with rate $\lambda$, then the expression $\exp(-\lambda t)$ is the probability of observing no counts in time interval $t$. This can be interpreted (after normalization) as the probability distribution (exponential distribution) of time intervals during which no counts are observed.
It is often said that this can also be interpreted as the distribution of times between counts. Although I'm sure this is right, I disagree: the distribution of time intervals in which no counts are observed is different than the distribution of time intervals between counts.
I would appreciate an explanation. Here's my reasoning.
I agree the exponential distribution is the time distribution for the first arrival, since the time origin (your stopwatch) was started at a non-specific time. However, the second arrival time must be measured with respect to the first one, which is a known time.
Take for example a low count rate (say 5 per minute) and a short time interval (0.1 sec). It is unlikely that I will get a count in that interval (i.e. probability of observing no counts is high -> exponential distribution). But that does not mean that 0.1 sec is a highly probable time interval between successive counts (quite the contrary actually). In fact, the exponential distribution says that the most likely interval is 0.
If you start your stopwatch at the time you record the first count, then the time interval before you record a second count would follow the distribution $P(1)= (\lambda t) \exp(-\lambda t)$, which seems to me to be the right inter-arrival time distribution.