Generate a binary number, using coin toss. Until you receive a predefined sequence. What is the probability that the number is a multiple of some k.

For example, the terminating sequence could be '11', and what is the probability that the number is a multiple of 3.

I understand questions like this can be answered by generating the whole transition matrix, and then solve like a markov chain. Is there a simpler rule which can give the probability for these type of questions?

Like the Conway's Algorithm in Penney's game;

http://plus.maths.org/content/os/issue55/features/nishiyama/index