It is the purpose of this paper to develop an expression for the probability of $x$ successes in $n$ trials, $P(n, x)$, where the probability of success on a single trial depends both on the number of the trial and on the number of previous successes. This result should prove useful in obtaining various probability functions. It will be noted that this work includes the case considered by Woodbury .
Ruric E. Wheeler. "A Variable Probability Distribution Function." Ann. Math. Statist. 27 (1) 196 - 199, March, 1956. https://doi.org/10.1214/aoms/1177728358