Probability of communication error

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Shouldn't the definition of the probability of communication error be:

 

instead of:

 

This way, the total probability mass of generating a message x and receiving a different message y is calculated. What is currently written, calculates the probability of error that is largest for a particular message. However, this does not take into account the fact that the probability of generating this message could be very small.

Where can I find the proof of the "Alternative formulation"?

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$SUBJ — Preceding unsigned comment added by Lockywolf (talkcontribs) 11:26, 30 September 2015 (UTC)Reply