NON-martingale approach to ABRACADABRA problem

One approach is through the theory of generating functions applied to finite automata, as described in section I.4.2 of Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick. (The book is available for free in pdf format online.)

Given a pattern $p_1 p_2 \cdots p_k$, we are interested in the number of strings which do or do not contain the pattern. To this end, we define the "autocorrelation vector" of the pattern as the vector of bits $c=(c_0, c_1, \dots , c_k)$ with $c_i = [[p_{i+1} p_{i+2} \cdots p_k = p_1 p_2 \cdots p_{k-i}]]$, where the brackets are Iverson's brackets. Then we define the "autocorrelation polynomial" of the pattern as $$c(z) = \sum_{j=0}^{k-1} c_j z^j$$ For example, the autocorrelation polynomial of ABRACADABRA is $c(z) = 1 + z^7 + z^{10}$.

It can be shown that the ordinary generating function of words not containing the pattern is $$S(z) = \frac{c(z)}{z^k +(1-mz)c(z)}$$ where $m$ is the cardinality of the alphabet, so for the pattern ABRACADABRA the generating function is $$S(z) = \frac{1+z^7+z^{10}}{z^{11}+(1-26z)(1+z^7+z^{10})}$$

The average wait time to the first occurrence of the pattern in a random string is $S(1/m) = m^k \; c(1/m)$.

For the pattern ABRACADABRA, with an alphabet of $m=26$ letters, the average wait time to the first occurrence is $$S(1/m) = 26^{11}+26^4+26$$