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Suppose you have a 26-sided die, each face is labelled from A-Z, what is the expected number of steps to observe the sequence "ABRACADABRA" for the first time?

ANS = $26^{11} + 26^4 + 26$

A common technique to handle these kind of problems is to draw a markov chain and state which correspond to observing the pattern "ABRACADABRA" is marked absorbing then you calculate expected number of steps till absorption, but I think this method is tedious (especially for the given sequence) and I believe there is some technique related to Optional Stopping Theorem that could be applied here, any help is appreciated.

Harsh
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  • See this question for a discussion of Conway's method for solving such problems. – lulu Mar 30 '24 at 12:43
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    An aproach using generating functions is shown here: https://math.stackexchange.com/questions/2961805/non-martingale-approach-to-abracadabra-problem/2962758#2962758 – awkward Mar 30 '24 at 13:19

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