[carstimon]

You pointed out one error in your own answer, here's another. I haven't figured out the adjustments, but presumably they cancel.

You've done

∫₀¹ ∫₀¹⁻ᶜ (2π r)³ dr dc

However,

- the `dr` integral is assuming that the radii are uniformly likely in [0, r]

- the `dc` integral is assuming that the centers are uniformly likely in [0, c]

You need to wait these integrals by the conditional probability distributions.

[carstimon]

Sorry, this was incorrect. In this integral we're just calculating the geometrical question, what is is the 5d volume of the collection of 3 points whose circumcenter is on {(x, 0) : x in [0, 1]}. So the "overall" probability of the problem has nothing to do with it.