Intuitive explanation of why entropy maximizes in a uniform distribution?

by roland 2y23rd Sep 20171 min read7 comments

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What is the best mathematical, intuitive explanation of why entropy maximizes in a uniform distribution? I'm looking for a short proof using the most elementary mathematics possible.

Please no explanation like "because entropy was designed in this way", etc...

https://en.wikipedia.org/wiki/Entropy_%28information_theory%29#Definition