I attempted to solve the April 2024 Jane Street puzzle, but I made some bad assumptions and did not get the correct answer.

I submitted 0.15889395849073286421 based on this computation in Mathematica:

In[1]:= N[Probability[
  Sqrt[(Cos[ptheta - diff] - Cos[ptheta])^2 + (Sin[ptheta - diff] - Sin[ptheta])^2] < Abs[(Sqrt[1/2] - pradius)/pradius]
  ,
  {
   Distributed[pradius, TransformedDistribution[Sqrt[u], Distributed[u, UniformDistribution[]]]],
   Distributed[ptheta, TransformedDistribution[2 Pi u, Distributed[u, UniformDistribution[]]]],
   Distributed[diff, TransformedDistribution[2 Pi u, Distributed[u, UniformDistribution[]]]]
   }
  ], 50]

Out[1]= 0.15889395849073286420654262653028487911805804181825

(takes ~30 seconds to run)

This is a bit too simple to be the answer, and indeed, the correct answer is 0.166186486474004.

Amusingly, in the course of writing the code, I found an input that ends up crashing the Mathematica kernel after ~30 seconds:

In[1]:= Probability[
 Sqrt[2 - 2 Cos[diff]] < Abs[(Sqrt[1/2] - Sqrt[pradius])/Sqrt[pradius]]
 ,
 {
  Distributed[pradius, UniformDistribution[]],
  Distributed[diff, TransformedDistribution[2 Pi u, Distributed[u, UniformDistribution[]]]]
  }
 ]

I am using Mathematica 13.2. I wonder what is going on under the hood.

Updated: