Consider the following scenario:
$\mathcal{A}$ and $\mathcal{B}$ play a game inside the unit disc $\mathcal{D}: $
- $\mathcal{A}$ chooses a point $p_0\in \mathcal{D}$.
- At step $n, \;\mathcal{B}$ then draws a line $\ell_n$ through $p_{n-1}$ and $\mathcal{A}$ chooses a point $p_n\in \mathcal{D}\cap \ell_{n}$.
$\mathcal{B}$ wins if he/she can force $(p_n) $ to converge, otherwise $\mathcal{A}$ wins.
$[$It turns out there exists a winning strategy for $\mathcal{B} $, which prompted the following question$]$
Suppose $\mathcal{A},\mathcal{B}$ play at random: what is the probability $\mathcal{B}$ wins?
$[$I strongly suspect this is $0$, but I am having a difficult time coming up with a justifiable reason$]$
Since player $A$ chooses the point randomly, the only way $B$ can win is if the line segments eventually get arbitrarily short. But if a line segment gets very short then $A$ is essentially choosing a fixed point on the boundary of the disc. Then when $B$ chooses the next random line, there is a positive probability that the line segment will be longer than some fixed positive length. Thus the probability that the line segments will stay very short forever starting at some iteration is 0. So player B loses with probability 1.