Main post link -> https://brilliant.org/mathematics-problem/2013c13/?group=YUgoPeVAfUIe
If you look at the written solutions, most of them count the probability as positive outcomes / total outcomes, and use the Hockey stick theorem to interpret the numerator.
Just as the technique of Choosing Correct Variables allows us to simplify the problem, the approach of finding a different perspective can allow us to cut to the heart of the matter. What is a possible different interpretation in this problem? Can you find a one-line solution to this problem? (I'd allow up to 4 short lines.)
Avi has an insight, and states that
By experimentation with smaller numbers of urns \(n\) and balls \(b\), the number of urns doesn't matter, as long as \( n > b \).
How do we prove this insight, without having to calculate the number of outcomes all over again? What is this probability?