A direct method to obtain the joint distribution of successes, failures and patterns in enumeration problems

Let E be a pattern (single or composite) defined over a sequence of Bernoulli trials. Let Xn, Sn and Fn be the number of occurrences of E, the number of successes and the number of failures in n trials, respectively. Also let Tr, STr and FTr be the waiting time for the rth occurrence of E in an infinite sequence of trials, the number of successes and the number of failures observed at that time, respectively. In this article we present some simple results revealing that, if we know either the distribution of Xn or the distribution of Tr we can instantly derive the joint distribution of (Xn,Sn,Fn) and (Tr,STr,FTr).