Distributions of periods and frequencies of runs in random binary sequences
The paper concerns random binary sequences in which 1's and 0's occur independently, with respective probabilities p and \(q=1-p\). As the main result it is shown that the probabilities p(k) that the length of the interval from the beginning of one run of 1's to the next is exactly k units, is equal to \((p^ kq-q^ kp)/(p-q)\) for \(p\neq q\) and \((k- 1)2^{-k}\) if \(p=q=1/2\). This result seems to be well-known for a long time. It could be a good exercise for students in a first course of probability.
scientific article; zbMATH DE number 4151502
scientific article