Was it not the case that people would have assigned very low probability to Fermat's last theorem too, at least before Wile's first version of proof? But then with Wile's first version, they would have assigned very large probability (1-epsilon, may be).

I am confused as to what benefits do we derive from assigning such probabilities? I am certainly missing something big here.

Why do you think so? As I understand it, most mathematicians thought Fermat probably correct.