I apologize for I think I wasn't entirely clear.
I agree with you there, if you're going to get called, going all-in with AA is correct.
The same with getting 4/5 chance of doubling your money, why wouldn't you?
You're also right about the 1/7 being a lot closer to 15% and a lot more accurate. I was trying to illustrate an important point and I felt 1/5 was easier to grasp in concept.
My point here is what most people call "bankroll management." There's a reason we don't invest it all in one hand/session, because 1/7 isn't that unlikely.
I believe that's why Poker Strategy recommends the "short stack" strategy for NL, and why it's so ingenious. Aside from the fact you may be called more easily from the opponent's standpoint, it protects you from one important factor: losing too much money on the less likely but very possible.
If bad runs rarely happens, we wouldn't have coined a name for it. Nor would poker professionals spend so much time finding other sources of income.
Back to my house analogy:
You would not invest your entire house into a 1/7 chance of losing all your money, but if you could do it multiple times, why not mortgage your house then divide it into 7 pots, and then try 7 times?
You must never invest all your money into one egg. Unless you're the luckiest person in the world (in which case, you can just go all-in with nearly every hand) you will lose eventually.
As far as getting beaten by a flush, the probability of someone drawing a flush by the river is roughly 6-7%.
If anyone still has trouble understanding, here's the summary:
Poker Strategy is winning in the long run. Odds are used as a guideline for what is likely to happen if repeated many times. Which is why it is important to play poker in a long-run strategy, not a 'bet everything and hope to win' strategy. This does not mean "don't go all-in" but rather "don't spend your entire bankroll on a few hands."