I find this exercise somewhere
The distance between two big fissures on a highway have an exponential distribution with mean $5$ miles. What is the probability that in a section of the road of $10$ miles will be two fissures?
Answer: $0.8647$.
If $X\sim\rm Exp(5)$ then the value of the proposed answer coincides with $\Pr[X\le 10]$. However it is not clear to me how this answer is achieved (I suppose that some assumptions are needed to achieve this answer).
To me $\Pr[X\le 10]$ means "the probability that the distance between two fissures will be at most $10$ miles", but how this statement can imply that in a stretch of $10$ miles the probability of (at least) two fissures will be the same? What kind of extra assumptions we need to show this implication?
An assumption could be supposing an underlying homogeneous Poisson process of parameter $\lambda$, then we reach the desired implication, but to me it is not clear that we can make this assumption easily from the text of the exercise.
My questions are two:
The answer of the exercise, without any extra assumption, make sense?
If not, what are the minimal number of extra assumptions we need to arrive at the proposed solution?