It's the same faith that people put into God. It's a blind faith, as you said, and sometimes it's really hard to keep believing..
But Gandalf and the other Istari were like prophets God sent...and Gandalf was in a sense even a picture of Christ himself, though not entirely. (There were not any perfect pictures of Christ in the LotR.) They came and though they could not see the end of all things, they did know that in the end, God's will would be done, somehow or other. Gandalf didn't know if Frodo was going to be the one who ended up accomplishing the task he set out to do, but he knew that someone was going to do it, because his job was to see it done, and Eru wouldn't have sent him on an impossible task...therefore, the quest couldn't fail.
Gandalf, Galadrial, and Elrond and all the rest who were wise didn't rely in sheer chance. Gandalf even talked about the greater power, in the book it's in Shadow of the Past, and in the movie it's in Moria, and because I don't have the book with me, I'm not going to quote because I don't have it memorized exactly, but he did mention that there was a greater power than evil and all things that happened were meant to happen.
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A young man who wishes to remain a sound atheist cannot be too careful of his reading. - C.S. Lewis
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