It is impossible to fully implement Bayes’ Theorem in all but the simplest circumstances, so we must settle for approximations. The key in most things is not to be exact but to avoid mistakes that can lead to large errors. Putting too much mental effort towards knowing the math behind a situation can be as damaging as putting forward too little effort as it has high opportunity cost and detracts from other aspects of your game or even your life. If you keep examining the trees, you may never see forest.
Jennifer Harmon has famously admitted that she knows little poker math, but she is successful because she focuses on other aspects while knowing enough of the principles to avoid making costly mathematical mistakes.
The most important mathematics to know are the odds of a given hand beating another given hand, or of a given hand improving. There are easy rules of thumb for approximating both in Hold ‘Em. Omaha can make the math more complex, to the point where calculations become impractical.
This first post will concentrate on a special case where the odds are both easy to learn and important to remember: Heads up hands in Texas Hold ‘Em before the flop. While these numbers tell far from the whole story when players are seeing flops and stacks are deep, if you play online there’s a good chance you will spend a large amount of your time playing in tournaments. Due to the need to finish these tournaments quickly, often the blinds will be so large that players will be forced to go All-In before the flop on a regular basis. This leads to large errors and large opportunities, which is why this is one of the best places to extract money at lower levels. Any examination of such events must begin with these odds, and even when stacks are reasonably large getting All-In before the flop is still common.
A good odds calculator can be found here and there are several others if you dislike the form this one takes. Playing around with it is highly recommended.
Here are the rules of thumb that I use:
An overpair, meaning a higher pair playing against two lower cards, is a 4:1 favorite (p=.8) to win against a lower pair. Excellent drawing hands have a slightly better chance than that, and non-drawing non-pairs are worse. A hand with a card matching the pair (such as AA vs. AK) is much worse, with p>.9 that the
A dominated hand, meaning an unpaired hand that shares the higher card with another unpaired hand but that has a higher kicker, is up to about 3:1 to win (75%) if the cards are high enough, with the edge decreasing if the kickers are both small down to about 2:1.
Two overs, meaning two cards higher than both your opponents’ cards, are generally a little less than 2:1 to win (p ~ 0.65), depending on the exact cards involved. Once again higher cards increase the edge a little. Having only one over is worse than having two, meaning you have the highest and lowest card, but it’s not a big drop (p ~ 0.6) and increases the chance your opponent has the only hand with straight possibilities.
Having two suited cards increases your chances of winning against one other player by about 3%. Having connected cards varies but is similar.
Very low cards, especially 4 or lower, weaken your hand somewhat because pairs you make with them are more likely not to matter, but this alone can’t cripple a hand.
The hand priorities change when there can be no more action after the flop. What matters most is dominating the opponents’ hand and avoiding being dominated in return. Being on the wrong side of a 3:1 or 4:1 advantage is a big problem, whereas being on the wrong end of a 3:2 edge because your opponent has a higher card than you is not so bad, and often the odds you are being offered are much better than that. The key is to make sure you have what are called live cards, meaning cards the opponent does not have. For this reason, although it is frequently necessary betting aggressively with an Ace that has a poor kicker is highly dangerous. You are likely to have the best hand, but the hands that are most likely to call you are those most likely to have you dominated and it’s very hard to have your opponent dominated.
A future post will discuss the implications of this. This is where math can take over the game, and consequently where players familiar with the game make the biggest and most obvious mistakes, mistakes which are often directly attributable to bias.
Rational Poker 
I use a freeware Windows program called PokerStove. It is available at: http://www.pokerstove.com/
Of note, it has the ability let me specify a range of opponents hole cards in addition to specific cards. For example:
QdQc
AA-77, AKs-AJs, AKo
random
10%
This is very convenient since I can rarely put my opponents on specific hands in specific situations. But I can often put them on specific distributions, like the examples above.
Examples would be really nice.
Here is a mistake I see quite often in the no-limit tournaments here in Vegas. (There are dozens of these low entry fee tournaments every day. The number of players typically range from 20-100. I like them because they only take 3-6 hours to play.)
The rake for these tournaments varies greatly, but let’s assume it’s 25%. Let’s also assume you tip the dealers 10% of any winnings. (Good tips are their only chance at middle-class wages!)
What does this mean? It means I try and avoid all-in situations early in the tournament where my winning advantage is less than about 35%.
Specifically, let’s assume I get a peek at my opponents hole cards and he has AK. He goes all-in. I fold my QQ, since my advantage is only 50-something vs. 40-something. (I usually assume the changes of a lower pair are roughly countered by the chance of a higher pair.)
You may wonder that since big-advantages are rare during a short tournament, how can I make any profit? Answer: generate folding equity. I have never been knocked out of a tournament where my opponents all folded.
The poker calculator you linked to is interesting, but not enough for a newbie to use when learning the odds.
I would find a lookup-chart, cheat sheet or equivalent, showing all of the required calculations, with shortcuts and pre-calculated results for common situations, quite useful
Um, what about priors? Where do they feed into these calculations?
E.g., that our opponent has an Ace pair, it’s something like (4 over 2)/(50 over 2) = 6/1225 = 0.0049… but since he’s playing very aggressively, we should somehow update this to a higher number, pulling it out of thin air?
Where would a beginner go to learn the terminology used in this post? Also, is there a quick start guide to poker odds that a beginner can use to learn the very basic math behind calculating odds – so as to check the math behind your numbers and figure other related ones on his own?