What is the average distance between two randomly chosen points in a square with a side length 1? The video presents how to estimate the answer and also derives the exact answer using integrals. Per suggestions in the comments, I have modified the title to indicate the difficulty level is VERY HARD (a college level calculus course and probability theory course are necessary background).
Sources and further reading
Difference of two uniform variables
Difference of two uniform variables and solving the double integral
Solving the double integral
Mean distance polar coordinates change
Integral secant cubed
Mean distance square
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