Why Does The Probabilistic Abacus Work?

In the article "The Probabilistic Abacus", by A. Engel (Educational Studies in Mathematics 6, 1975, p. 1 - 22), we have introduced the probabilistic abacus and we have applied it to absorbing Markov chains. We have explained in detail how to compute absorption probabilities and expected times to absorption, but we gave no proofs. In this paper we give more applications of the abacus and we supply proofs for most of its properties.