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Before we continue the interesting conversation, we need to clarify that I can detect a pitfall in the way you are stating the problem.
Your { P1, P2, .., Pn} is not a probability distribution and this is of course why the tuple does not add up to 1 but to 2. The logical reason of course is that Pi is not independent from Pj.
I will continue later with a more detailed view
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whereof one cannot speak thereof one must be silent
Ludwig Wittgenstein
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