Yes. 2.5% is what many people have determined by using the 3000 fate achievement to count fates.
Which means on average you should get it within 40 fates but it wouldn't be unlikely for it to take 80 or even 120. 40 fates is about 2 hours if you're speed running them (not staying for them to finish). If you stay for them to finish then it's much longer.
Speed running them is the key. Do not stay for them to finish. it's completely unnecessary. Even if you think you need gold, it takes little effort to get gold and the fate doesn't need you to finish. You're only making the grind several times worse than it needs to be.