On random split of the segment
We consider a partition of the interval $[0,1]$ by two partition procedures. In the first a chosen piece of $[0,1]$ is split into halves, in the second it is split by uniformly distributed points. Initially, the interval $[0,1]$ is divided either into halves or by a uniformly distributed random variable. Next a piece to be split is chosen either with probability equal to its length or each piece is chosen with equal probability, and then the chosen piece is split by one of the above procedures. These actions are repeated indefinitely. We investigate the probability distribution of the lengths of the consecutive pieces after $n$ splits.