If $N$ points on the circumference of a circle are chosen at random, what is the probability $F(\theta)$ that the maximum gap between neighboring points is at least $\theta$? Because the gaps sum to $2\pi$, the maximum must be at least $2\pi/N$, so $$F(\theta)=1 \text{ for } \theta\le\frac{2\pi}{N}.$$ At the other extreme, the solution to this problem shows that $$F(\theta) = N\left(1 - \frac{\theta}{2\pi}\right)^{N-1} \text{ for } \theta\ge\pi.$$ Is there a closed-form solution for any other values of $\theta$?
Update: The general closed-form solution is given in the answer below. In terms of the notation in the question, it is $$ F(\theta) = \sum_{k=1}^{K} (-1)^{k-1} {N\choose k} \left(1 - \frac{k \theta}{2\pi}\right)^{N-1}, $$ where $K = \min(N, \lfloor{2\pi/\theta}\rfloor)$. This reduces to the limiting cases given in the question when $\theta \le 2\pi/N$ (in which case $K=N$) and when $\theta \ge \pi$ (in which case $K=1$).
