Yayy, my first post!
The Theory of Poker:
If you played a million hands with someone heads up, or a million hands at a 9 person table, the only way you make money in the long run is if other people make mistakes, or rather, make more mistakes than you do. Mistakes include folding with the best hand, calling with a worse hand (without correct odds), or even making a less positive EV bet (like calling when raising is optimal). In the end, if you make less mistakes than the other person, you win.
Intro Poker Probability:
To talk about poker, we need to talk about probability first. If there is 12 dollars in the pot, and you face a bet of 2 on the turn (with one card left to go), and you have the nut (best) flush draw and you know your opponent has Aces (a pair of Aces), you should actually call. There are 9 outs (cards that will allow you to win the hand) in the deck, so you need 37:9* odds, or a little over 4:1, and you are getting 6:1, so your call has positive EV. But poker is never as simple as this since 95+% of your decisions are not on the river.
Conditional Probability and Implied Odds/ Reverse Implied Odds:
Conditional Probability is hugely important in poker. Ask yourself how likely you are to be best, but also ask yourself what can happen if you are in fact best on future streets, and what can happen if you are worse on future streets. So here's an ex. You raise one off the button (button is the dealer, and 1 off means you're to the right of the dealer and there are 3 people left to act. 3 off would mean 3 seats to the right of the dealer and there are 5 people left to act) with A3s (Ace 3 suited) and the dealer and BB (big blind) call. Flop is A 7 4 rainbow (no suit matches), so it's great to think your pair of Aces are best, so you want to bet right? Well, you can bet and take down the pot most of the time, but you're only getting called with a better Ace (Ace and a higher kicker/other card) and only a few worse hands. Let's think about checking (not betting). You're likely to win the hand at showdown (after all cards are revealed), unless someone hits 2 pair or runner - runner something. And if the guy behind you has a better Ace and bets, at least you have more information to make a better decision. So in fact, checking is best here, because of conditional probability.
You have pocket fives and there's an open raise (the first raise and first action preflop), call and you're on the button (you're the dealer). So the pot is just above 2:1, and the chance you flop a set (another 5 comes out) is 8/9:1 against. This sounds terrible, but when you do hit your set, you may win much more than what's in the pot in later streets. You have a hidden hand and top pair is likely to give up a lot of money in that scenario, so you have implied odds. As a general rule, the other players need about 20x the bet you're calling pre-flop behind (meaning if the raise is to $7, they need about $140 behind to make your call worth it (to make up for times they fold to you and you don't get paid).
Reverse implied odds is the opposite. You open raise with AJ and a guy re-raises you preflop. The pot odds let's say are 2:1, and let's say he's reraising sometimes with worse hands like KQ, AT (Ace-Ten), and if he's got a pocket pair like T's or 9's it's a coin flip, so you should call right? Well no. If the guy has AK or AQ, you're absolutely crushed and you don't know it. If an Ace comes out, you're likely to lose a lot of money. If the guy has A's, K's or Q's and the flop comes out J high, you're losing even more money.
I'll end this post here, because it's getting long. But I promise I'll get into more interesting stuff down the road, and how poker is good simulation for other things. Most importantly, I can teach you how to be positive EV at the casino.
*with a 1/3 prob of winning, you need at least 2:1 odds, odds are listed as failures:successes, unless you say 3:2 "favorite", in which case it's successes:faiures. It's not very intuitive for us math geeks, but actually gamblers use this because it frames the probability correctly, if you have 2 in the pot and face a bet of 1, you need 2:1 odds.