how the hell did you find out what the chances are?

Who? Herbert explained quite clearly how he did it. If you aren't familiar with the terms, look them up.

http://en.wikipedia.org/wiki/Markov_chain
If you can't understand something, read the relevant articles first.

The way I did it wasn't as elegant. The state after any number of solves can be represented with two natural numbers:

*n*, the length of the current streak, and

*m*, the length of the longest streak so far. Doing one solve gets us from (n,m) to (0,m) with probability 5/6, and to (n+1,max(n+1,m)) with probability 1/6. At the beginning, we are obviously at state (0,0) with probability 1. Keeping track of all different states and the distribution is fairly easy.

Here's the code I used, it takes a few seconds to run.

im repeating high school algebra 1 so i have no idea what you all are talking about.

im reapeating algebra 1 next year

So what? Are you not allowed to learn outside school?