1/2^n, I believe (where n is the number of lines after the first one, otherwise 1/2^n+1). So halved every time: 100% with 1 line, 50% with 2 lines, 25% with 3 lines, and so on. Width is irrelevant (you can do this in your head comparing w=1 n=2 to w=2 n=2 for example).
Also, I don't believe it doesn't depend on width. I think with constant height, the probability of completable one should increase with width, and become almost 1 for very large widths.
Hm, good points, looks like I need more sleep. Looking at the video again, a path is valid if two slashes follow each other, with the same pattern shifted by 1 the line below.
But I wasn't treating /\ as an invalid on the first line. Eg thinking of:
/\/\/\/\
\\//////
/\\\\\\\ … and so on
(Using the /\ pattern, paths can go back up and come back down, so this requires a lot more thought than I instinctively put into it)
Expressed in terms of line width (w) and number of lines (n).
[0] Where there's a valid path from the first line to the last.