We are given R (radius of circle), u (speed of person) and t (time traveled by person). Hence, we can calculate the distance traveled, D.
Assume that the starting position is evenly distributed in the circle.
Also, the person can take any direction, and the direction will be steady.
Is it possible for this to be solved for R,u and t? I can only find a solution which includes D_start, the initial distance of the person from the center of the circle.
Fix the disk radius $R=1$ without loss of generality. I selected the speed $u=1$—traveling a unit distance in $1$ second—just as an example. Then, on average, it takes about $0.85$ seconds for a random start point, aiming in a random direction, to exit the disk. So for $u=1,\,t=0.85$, the probability of exiting the disk is $50$%.
$50$ random unit-length rays within a unit-radius disk.
For an arbitrary speed $u>0$, it takes about $0.85/u$ on average to escape the disk. However, I do not know an analytical expression for the $0.85$ constant.