Probability of any random n points on a line being within a given distance

Given a virtual line segment in cm and n random points on it, what is the probability that the distance between any consecutive points on the line is less than a given minimum distance?

For Example:
n = 10 points
lineSegment = 1000 cm
minimumDistance = 2 cm

Running a Montecarlo simulation I took the following steps:

  1. generate n random points
  2. sort the points by order of smaller first
  3. calculate the distances between consecutive points
  4. count how many distances are smaller or equal to the minimumDistance.

Link to Python Montecarlo simulation on replit:

Is there an analytical solution to deal with any n number of points?
If yes, then I need:
1. Mathematical proof/explanation
2. algorithm/formula


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