Are there any connections between the probability of poisson distribution when x=k and x=k+1?

Question

We know in poisson distribution, we have: $$P(x=k)=\frac{\lambda^k}{k!}e^{-\lambda}$$ so $$\frac{P(x=k+1)}{P(x=k)}=\frac{\lambda}{k+1}$$ I want to know are there any intuitive explanation of this result?

Answer 1

Dividing two probabilities does not, in general, have any meaning.

Note that this ratio tends towards 0, all this tells you is that RATIO decreases.

The ratio shouldn't be surprising given the parameter $\lambda$