Suppose $5$ blue points and $5$ red points are selected in the interval $[0,1]$. What is the probability that the points will interleave each other?
Interleave as in one blue point followed by one red point and so on or one red point followed by a blue point and so on.
Any tips for solving such problems is highly appreciated. Thanks in advance.
I would start by trying to rephrase the problem as a "standard" combinatorics problem. For instance, notice that there is no difference between choosing colored dots on the unit interval and choosing an ordering of *****|||||.
There are $\frac{10!}{5!5!} = 252$ orderings and two of them are interleavings, so the interleaving probability is $\frac{1}{126}$.
Imagine instead selecting the $10$ points first, then colouring $5$ red and $5$ blue, chosen at random.
There are $2$ choices that give interleaving, and $\binom{10}{5}$ ways to choose the red points. So the probability is $\frac{2}{\binom{10}{5}}$.
