This may be a silly question, but if we look at Poisson distribution at $k = \mu$ and $k = \mu-1$, for integer $ \mu $ the values are nearly identical.
from scipy.stats import poisson
poisson.pmf(k=5, mu=5)
Out[39]: 0.17546736976785068
poisson.pmf(k=4, mu=5)
Out[40]: 0.17546736976785063
Should this be interpreted as arrival of 4 events has similar probability as arrival of 5 events given a common rate of 5 events per period or population? This breaks my intuition, why so?