Google Earth circles probability

Hi everyone. Not sure if this is the right forum but I'll give it a go.

Q: If you draw a circle on a sphere, what are the odds that a randomly chosen point will be within X km of that circle?

I'm happy to into more detail of what I think the solution is if this is the appropriate place to discuss this. I just want some more minds to bounce it off to show me where I'm astray (or not).

Thank you.


TS Contributor
I second Dason's question. You need to be careful to specify the distribution of the points on the spherical surface (?) or otherwise you may easily run into a similar trouble from Bertrand's paradox. Also you need to be careful in your hyperbolic geometry calculation, although this case should be the simplest one.