What it seems like to me is that before ~900-1200 pulls you chance for SR is 1 in 100,000, maybe less. Then I suspect after you "trigger" the SR unlock, your chance becomes the same as any other rare, until you get a SR.

This is how I think the mechanics works. For every 10 you pull, there is 1 guaranteed "good" pull which will reward a ninja from your current "pool" on 10-50 this ninja is from the non-rare pool. on pull 60 it is from the "rare" pool. Which of the ninjas from the pool you get is completely random. Once you have >900 pulls, then the SR is added to that rare pool. Those SR have an equal chance of being chosen as any other rare, but it seems like each time you "miss" the SR it gets added to your pool again. So, there are 10 Rare ninjas in GNW and 2 SR. So you chances at first are 2 in 12, then you r next 60 pull your chance is 4 in 14, 6 in 16, 8 in 18, once you get to ~1200, your chances are better than 50/50, but still not guaranteed. Once you get your first SR, then those ninja drop out of your rare pool again until you reach 1800 pulls. But, on every pull your other 9 cards have that super low chance of giving you a SR or the hashirama frags (which seem to be a lot higher chance than the SR).

Of course this is all just my theory, I could be completely wrong on the mechanics of it, but this seems like the most logical to me.