These probabilities are not required to sum to 1, because they are not incompatible and exhaustive possible outcomes of an experiment. More obvious example to illustrate:
P(6-sided die coming up as 6 | today is Monday) = 1/6
P(6-sided die coming up as 6 | today is not Monday) = 1/6
1/6 + 1/6 != 1
I think your example is not suitable for situation above - there I can see only two possible outcomes: X happen or X not happen. We don't know anything more about X. And P(X|A) + P(X|~A) = 1, isn't so?
It's not obvious that one is better off with the truth. Assume that for some desirable thing X:
P(X|I believe X will happen) = 49%
P(X|I believe X won't happen) = 1%
It seems I can't rationally believe that X will happen. Perhaps I would be better off being deluded about it.
Sorry, I don't understand - why does sum of probabilities not equal 100% in your example? Assume that you missed "5" in "P(X|I believe X won't happen) = 1%"
Perhaps I would be better off being deluded about it.
But for what reason?
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I think that great example of growth mindset is "Baby steps" anime and manga (http://mangafox.me/manga/baby_steps/). Boy become proffesional tennis player from scratch using deliberate practice and hard work.