Play $(n+1)t$ rounds of the same coin-tossing game and the coin is fair ($n$ is a fixed natural number). Please help me find the following probability:
$P$(the number of rounds of tossing that show head is no less than $t$ after all the tossing is done, and at any time point during, the number of coin landing head is no less than $1/n$ of that of tail).
If possible, please also show that when $t \to \infty$, this probability becomes/does not become $0$.
What I think: If n=1, it is similar to the Catalan number. In this more general case, I can calculate the probability of the number of head landings being x after all the tossing, with x between 0 and (n+1)t. Aggregate this on the interval [t,(n+1)t] can get the probability without the restriction of 'at any time point during ... no less than 1/n of that of tail'. Basically I followed the approach used in here. Between page 473 & 475 (theorem 12.1 till 12.3). But I do not know how to deal with the restriction, neither do I know how to evaluate the limit of this probability.