FWIW, unless you have reason otherwise (you may very well think some Fs are more likely than others), thereβs some symmetry here between any function F and the function 1-F, and I think if you apply it, you could say P(F > 1β2) = P(1-F < 1β2) = P(F < 1β2), so P(F < 1β2) β€ 1β2, and strictly less iff P(F = 1β2) > 0.
If you can rule out P(F = 1β2) > 0 (say by an additional assumption), or the bet were on F β€ 1β2 instead of F < 1β2, then the probability would just be 1β2.
FWIW, unless you have reason otherwise (you may very well think some Fs are more likely than others), thereβs some symmetry here between any function F and the function 1-F, and I think if you apply it, you could say P(F > 1β2) = P(1-F < 1β2) = P(F < 1β2), so P(F < 1β2) β€ 1β2, and strictly less iff P(F = 1β2) > 0.
If you can rule out P(F = 1β2) > 0 (say by an additional assumption), or the bet were on F β€ 1β2 instead of F < 1β2, then the probability would just be 1β2.