Li's Martingale Approach to Pattern Waiting Times

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Martingales / Li's Theorem.

Your Task
A fair 26-letter keyboard generates characters uniformly and independently at random ($1/26$ each). Let $T$ be the stopping time when the pattern $\text{ABRACADABRA}$ first appears. Using Li's martingale gambling approach, derive the exact expected waiting time $\mathbb{E}[T]$.

Constraints

• Transitions and increments satisfy the Markov or martingale property with respect to natural filtration.
• Stopping times are almost surely finite.