[weak] information paradox

While (still!) looking at questions on Cross Validated on Saturday morning, just before going out for a chilly run in the park, I noticed an interesting question about a light bulb problem. Once you get the story out of the way, it boils down to the fact that, when comparing two binomial probabilities, p1 and p2, based on a Bernoulli sample of size n, and when selecting the MAP probability, having either n=2k-1 or n=2k observations lead to the same (frequentist) probability of making the right choice. The details are provided in my answers here and there. It is a rather simple combinatoric proof, once you have the starting identity [W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1, 1968, [II.8], eqn (8.6)]

{2k-1 \choose i-1} + {2k-1 \choose i} = {2k \choose i}

but I wonder if there exists a more statistical explanation to this weak information paradox…

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