[Galadriel55]

I think that rationally, Gandalf knew that there was very little chance of anyone from the Fellowship even getting to Sammath Naur, forget about destroying the Ring. At the same time, though, he might have had a feeling that at a certain point something would happen which was statistically improbable and unpredictable that would turn things around. Part of it was a faith and trust in Eru, but part of it was just accepting that even without external intervention, you can't control all circumstances but you have to do your best. I think the last part is the most important - regardless of what Gandalf could or could not foresee, he knew he had to try. As another lovely book says, "If you try and lose then it isn't your fault. But if you don't try and we lose, then it's all your fault." Gandalf just went with the best of his options. And it really wouldn't be his, or Frodo's, or anybody's fault if things turned out differently. They all tried their best, in their own way. That was as far and as much as Gandalf would really need to see to make the choice.