# Prove that entropy is maximized when probability is $1/n$

How can be proven that the entropy of a dice roll is maximized when the probability of each of its $6$ faces is equal, $1/6$?

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Suprise is defined as -log(p{X=x}). A good way to think of entropy is the "expected surprise". In this sense, its easy to see that the uniform distribution maximizes the expected surprise.

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not sure how to do it... can you help? – whynot Oct 26 '12 at 5:55

The entropy is given by $-\sum p_i\ln p_i$. Use Jensen's inequality with the logarithm function.

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not sure how to do it... can you help? – whynot Oct 26 '12 at 4:52