The first run preceded by a quota

Let a,b,c be posi tive integers wi t h b"5, min (a,c l ) . In an infin ite sequence of independen t trial s with different success probabilities, define a " run" to consist of a consecutive s uccesses, and a "q uo ta" to consist of c or fewer consecutil'e tria ls containing b or more successes . Recursion formulas are derived for the probabi li ties governing the first appearance of a run immediately preceded by a quota.