## A maths puzzle: Bertrand’s Paradox and “random chords”

I was reading the wikipedia page for Bertrand’s Paradox and was wondering about the various ways one can pick a “random” chord on a circle. I found this on the discussion page:

- pick a random point inside the circle and a random direction, make a chord in the direction through the point
- pick a random point inside the circle and a point on the circle, join them up and extend into a chord
- pick a random diameter and a distance away from the diameter, draw a chord parallel to the diameter at that distance. (there are 2 chords in each case, but allow the distance to be negative and then pick an “up” let positive distance mean take the chord “above” the diameter etc)
- pick a random diameter and a random point inside the circle, draw a chord through the point perpendicular to the diameter
- pick a random diameter and a random point on the circle… etc

As well as the 3 methods listed on the page itself, this is quite a selection of methods. So which are equivalent? I imagine that each of those reproduced above is equivalent to one of the three listed on the page, but I haven’t really thought about which are equivalent to which…

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