Way to rank binary sequence

Bear with me as I try to word this question well (I'm a mathematical modeler, but not a statistics guru).

We want to assess the response patterns of a binary timeseries. The data are from people answering questions for information they have not previously seen. The scenario is that answers are either correct or incorrect, and they have to get an answer correct twice in a row before it is "retired." Since people have not previously seen the information, the "perfect" pattern is 0-1-1, meaning the person answered incorrectly, then was presented the correct answer/information, then retained that for the next two attempts.

I'm trying to assess people's "trend" and am seeking a pointer to how this might be possible. It's not enough to collect responses, since two people having 3 correct responses and 3 incorrect responses might reflect very different "trends." To wit:

• Person A: 0-0-1-0-1-1
• Person B: 1-0-1-0-1-0

In the first case, the person experiences a trend toward understanding. In the second, the person is alternating with no trend toward understanding. Ideally, we'd come up with some number normalized to assess some distance from the perfect 0-1-1 pattern. For instance, 1-0-1-1 would be more distant than 0-0-1-1, but closer than 0-0-0-1-1. Even a distance from 0, 1, or some other arbitrary point would be good if it was internally consistent.

My first thought would be to look for something in signal analysis– assess the periodicity of the signal of responses. However, I was wondering if the statistics community has some manner to assess a binary sequence like this. Of course, I'm not looking for an easy answer (since there might not be one) but pointers in the right direction would be helpful.