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I had almost this exact same argument with my uncle, who has the same type of personality. He'd watched a bit of WPT on TV and very quickly formed the view that poker is complete luck.
I apologise in advance for the long post, but I think the conceptual argument goes something like this (feel free to expand on the points and add in your own examples as required!).
Let's start with a game that is indisputably complete luck. Say, roulette. No-one has any control over which number is coming up next (assuming an unbiased wheel), so whether you win or lose is utter dumb luck. BUT, the other thing we know about roulette is that over the long term, the house will ALWAYS win. Why? Because of course it makes sure that the odds are skewed in its favour. When your number comes up you win 35x your bet, but because the number will only come up once every 38 spins on average (or 37 if there is no 00), over the long term you WILL lose.
Poker is like roulette insofar as the hand you get and the hand your opponent gets is utter dumb luck. Whether you would win or lose a particular hand at showdown is utterly out of your control.
The key here though is that unlike roulette, the chances of winning at showdown vary wildly from hand to hand, depending on the cards. And, also unlike roulette, the size of the return you can get on your bet also varies wildly, depending on how much you bet and what the other players do, and doesn't necessarily bear much relationship to the chances of winning at showdown. It will sometimes be much better, and at other times much worse. So, a player who picks his spots well and only puts his money in when the potential return on his bet is better than his odds of winning (ie. consistently makes +EV decisions), will have the odds skewed in his favour and will win in the long term. So how do you make +EV decisions? This is where part of the skill comes in - you calculate odds and probabilities, and only stick your money in when you have the best of it. Simple.
So my uncle then points out that calculating odds and probabilities is well within the capabilities of many people, and if everyone is doing it, they can't all win because its a zero-sum game (actually less than zero due to the rake). So you can do that perfectly and still lose over the long term. Aha, I says, thats where the other key characteristic of poker comes in, which is that it is a game of incomplete information. If all hands were played face up, you could always mathematically calculate your chances of winning at showdown and therefore the return on your bet you would need. But the fact is that you can't see some or all of your opponent's cards, so except for the extremely rare case that you have the stone nuts (or anti-nuts), you don't know exactly what your chances of winning at showdown really are.
This is where the other more important elements of skill come in - player reading, hand reading, bet types, bet sizing, etc, etc. And it works in at least two ways. First, the better you can read your opponent through betting patterns, tells, etc, and accurately put him on a range or hand, the more accurate your calculations will be and therefore the more +EV your decisions. Secondly, even better, a very good player can use these skills to take the cards right out of it on occasion. If you understand your opponents' style and tendencies, you can exploit them to get more of his money in if he has the worst hand, or get him to fold the best hand. And of course thats why position is so powerful - it gives you that extra piece of information (your opponent's action ahead of you) to help you make your decision. Doyle Brunson famously said in Super System that if he could have the button on every hand, he could beat most any poker player without even looking at his cards. This is ALL skill.
I think the best way to think of all casino games is as a series of betting situations that are created through various random and artificial means. Poker, like blackjack, uses cards to create those situations, while roulette uses a ball and a big wheel. For most of these games the house makes sure that your potential return if you win is ALWAYS less than the odds of winning. The single difference with poker is that the chances of winning are not always worse than the return for winning, and it takes SKILL to recognise and exploit those times that it is not.
That's why poker is a game of skill.
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