Today I wrote a document with a draft for a book. That said we will continue our discussion on Ballot theorem and probability from simple random walk. Ballot theorem was originally used with urns. Suppose that there is an urn with a azure balls and b black balls totalling n balls. A sequence of a balls when drawn has a uniform  distribution. The ballot theorem originally proposed in 1887 posed a question this way. If there is such an urn and there is sampling without replacement what is the pro ability that the azure balls will lead the black ones for some positive finite and restricted draw. The problem was solved the same year and the probability was found to be (a-b)/n as long as a > b