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LawrenceC comments on Open thread, Dec. 21 - Dec. 27, 2015 - Less Wrong
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Wait, how would you get P(H) = 1?
Fine. p(H) = 0.5, p(H|A) = 0.2, p(H|B) = 0.15, p(H|C) = 0.15 It's not really relevant to the problem.
The relevance is that it's a really weird way to set up a problem. If P(H)=1 and P(H|A)=0.4 then it is necessarily the case that P(A)=0. If that's not immediately obvious to you, you may want to come back to this topic after sleeping on it.
Fair enough.
\sum_i p(H|i) need not add up to p(H) (or indeed to 1).
No, it doesn't.
Edit - I'm agreeing with you. Sorry if that wasn't clear.