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Most fast decks are built on a "curve", meaning there are more one drops than 2 drops, more 2's than 3's and so on. This doesn't mean they are lucky but rather that they are more likely to draw cheaper minions then expensive ones, and therefore are more likely to have better plays early in the game.
Aggro decks are built for early game consistency. Take a Genn Paladin deck, for instance; you're guaranteed a turn 1 play, and the deck might run as many as ten two-drops. A Baku deck, on the other hand, might have 8-10 turn 1 plays. It's just statistically likely that you're going to find what you need with the mulligan.
I think your analogy is a bit off. In order to end games that quickly, a well built aggro deck only needs to curve out, not necessarily draw an exact, specific series of cards. One of those things is much easier than the other. It's simply a matter of constructing your deck correctly (pretty much exactly as 1HuntAlone described above). Basically, a well tuned aggro deck will have multiple tools that are more or less interchangeable, allowing them to finish games quickly consistently. Yes, drawing the right cards early can speed up the process, but even with just average draws, they are looking to win well before turn 10. It's combo decks which tend to require specific pieces in order to close the game, and those decks (not surprisingly) tend to take much longer to finish things out on average.
And speaking of one thing being much easier than the other... the difference between a straight and a royal flush is astronomical. The odds of drawing a straight on the initial deal in a 5 card poker hand is roughly 250:1. A royal flush is just shy of 650,000:1. You referred to them in a way that made them seem almost interchangeable. I'm not implying that you are not aware of the difference, rather I just wanted to point it out for those who may not be all that familiar with poker. That's a mistake you don't want to make at the table.
Between being able to build your deck for early consistency and mulligan for the cards you want, something would be wrong if aggressive lists didn't assemble a high-pressure curve. I couldn't tell you the specific numbers, but the difference in probability between a royal flush (very improbable) and an aggressive list drawing on-curve threats (very probable) is enormous. It's what they're supposed to do. As the player on the back foot it's your job to respond.
The strength of aggressive play is that they ask the questions and there are no wrong questions. By contrast, there are wrong answers.
My games always go longer than 15 Turns and most of them end in fatigue.
The true is that consistency is the key to get the games you want, I tech against fast decks and so I almost always get to later turns, decks in the other side are built to curve well and have the answers to by pass defense and deal damage fast.
Also Royal Flush isn’t a right analogy, as here you have several strong plays with as little as half your deck, drawing 5 good cards from a pool of 15-20 on a deck of 30 isn’t that hard as your observations of HS games have shown you.
I love the game. I enjoy playing. I have 5 golden heroes and a decent card collection. However I have been playing long enough to remember when strategy played an integral part of the game. Sadly that is no longer. Most of the time the winning player has no cards left. This is because they simply play the cards as dealt on curve, empty their hand, and top deck the game winner. There is no thought involved.
I am not complaining about losing. I win as many mindless games as I lose. I just miss strategy.