Let $n \in \mathbb { N } ^ { * }$ and let $\gamma = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 n } \right)$ be a Dyck path of length $2 n$. For $t \in \mathbb { N }$, express $\mathbb { P } \left( A _ { t , \gamma } \right)$ as a function of $n$ and $p$, where $A _ { t , \gamma } = \bigcap _ { k = 1 } ^ { m } \left( X _ { t + k } = \gamma _ { k } \right)$.