Dartboard Rotation

Consider a thin, uniform circular dartboard of mass M and radius R, initially at rest, constrained to revolve about a fixed tangential axis only (but can revolve freely about that axis), as shown: A player hits a point dart (same mass M) on the dartboard with velocity 'v' perpendicular to its plane, which gets embedded in it (assume the shot is within the dartboard).

The points for the game are expressed as a percentage of the initial angular velocity attained by the embedded dartboard system to its maximum possible value ( $\omega_{max}$ ). (Example- if the dartboard attains initial angular velocity $\frac{\omega_{max}}{4}$ , then points gained = 25).

Now, if the probability of a player scoring k or more points in 1 shot is $\frac{arcsin(\frac{1}{a})+\frac{\sqrt{b}}{c}}{\pi}$ where a, b, c are integers, b and c are co-prime; find a + b + c.

Particulars:

k = $\frac{400\sqrt{5}}{9}$ (≈99.3808) points

R = 1 m

v = 1 m/s

This problem is original (up to my knowledge :)). Corrections for any ambiguities will be helpful.