Details

Snapshot 1: if the probability of moving from disk 1 to disk 2 is zero, the probability of winning is 1

Snapshot 2: if the probability of moving from disk 2 to disk 1 is zero, the probability of winning does not depend on the probability of continuing with disk 2

Snapshot 3: if we always continue with disk 1, the probability of winning is indeterminate because the game never ends

Snapshot 4: if we always continue with disk 2, the probability of winning is likewise indeterminate because it is possible that the game never ends

Snapshot 5: if we always move from disk 1 to disk 2 and from disk 2 to disk 1, the probability of winning is again indeterminate, because the game never ends

The Demonstration is based on [1, pp. 28, 244–247], where the probability of winning the spin game is calculated by a clever method (note the misprint in the boxed probability on p. 245: in the denominator should be ).

Reference

[1] P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton, NJ: Princeton University Press, 2008.