SCHWEITZER 2021 Problem 10

Consider a coin with a head toss probability $p$ where $0 <p <1$ is fixed. Toss the coin several times, the tosses should be independent of each other. Denote by $A_i$ the event that of the $i$ -th, $(i + 1)$ -th, $\ldots$ , the $(i+m-1)$ -th throws, exactly $T$ is the tail. For $T = 1$ , calculate the conditional probability $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$ , and for $T = 2$ , prove that $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$ has approximation in the form $a+ \tfrac{b}{m} + \mathcal{O}(p^m)$ as $m \to \infty$ .