As every self-respecting poker degenerate knows, there are 1,326 unique starting hands in the Texas Hold'em universe.

When you peek at your hole cards at the beginning of a hand, all of the 1,326 combinations are equally likely. Your chances of seeing [AhAs] are exactly the same as seeing [7d2c]. That is, the 1,326 starting hands are isometric with regard to probability: each has the same chance of occurring. So when you stare down at the opaque patterned backing of your hole cards prior to peeking at their undersides, you're not looking at a specific hand. You're looking at a probability cloud containing 1,326 possibilities, one of which will manifest when your hole cards are formally observed.

Prior to observation, a given player's hand isn't a hand; it's a probability cloud containing N distinct possibilities.

We call this probability cloud a hand distribution or a hand range and far from being a recondite mathematical theory of interest only to statisticians and poker geeks, it's one of the most powerful weapons we have in the battle against incomplete information. Good players use this weapon all the time, consciously or unconsciously. Every time a player deduces a piece of information, however vague, about an opponent's hand, he's creating a hand distribution.

My opponent has either Aces, Kings, Queens, or Ace-King. Not sure which. He's strong, though.

My opponent has two cards (any two cards) of a particular suit, for the flush. Or he might be bluffing the flush.

My opponent either has a set or an overpair.

My opponent has top pair with a good kicker.

Each of the above statements can be expressed precisely by assigning the opponent a distribution containing or more potential specific hands. For example, if the preflop action convinces you that your opponent has Aces, Kings, Queens, or Ace-King, you've assigned him a distribution—a probability cloud—containing 34 distinct possibilities, each of which is a specific two-card starting hand.

A distribution can be as small or as large as necessary to countenance all the possibilities. If you know an opponent's specific hole cards (perhaps you caught a glimpse of them), that opponent has a hand distribution containing a single hand. The probability of him having that hand is 1.0 or 100%. This is a truism, but a necessary one in order to normalize the underlying logic. Everything is a distribution. If everybody folds to you on the button and you're considering whether to raise, the small blind and the big blind (since their cards are completely unknown) have a random distribution containing every possible 2-card holding. After you raise, of course, and the small blind re-raises, you'll have to revise his distribution to take into account the new information. But you're always working in terms of distributions. Every player's hand at the table can and should be thought of as a distribution containing one or more hands.

Distributions have enormous utility because pinpointing an opponent's exact hole cards is difficult. Even when we're 95% certain our opponent has a given hand, there's always that pesky 5% chance he's splashing around with something else. And usually we won't be 95% certain; we'll be 75% certain, or 50% certain, or an unqualifiable amount of certain. And this is where one of the most common and egregious mistakes in poker is made, namely:

Playing in such a way as to maximize your value against the specific hand you believe your opponent has.

The thought process goes something like this:

Okay, I'm pretty sure my opponent has such-and-such a hand here. Yes. He raised preflop, I bet into him on the flop, now he's raising...yep, he's got such-and-such. I might be wrong here, but I'm gonna go with my read. I'm all-in / I fold / etc.

There's nothing wrong with making a read and sticking to it; the problem is that the above train of thought is usually indicative of a read that is too specific—far more specific than the available information warrants. When we put all our eggs into one basket by making a very precise, possibly incorrect deduction about an opponent's cards, and when we base our betting decisions on that deduction, we front-load the difficulty of poker onto our (flawed) ability to extract (incomplete) information from a (loosely-wired) poker situation.

In other words, we fall into the trap of deterministic strategy. We play as if our opponent had a specific hand.

But until we actually observe an opponent's hand for ourselves, it's not a hand, it's a probablistic distribution of potential hands. Sometimes this distribution will coalesce, based on the available evidence, and we'll be able to say with confidence that the opponent is holding such-and-such a specific hand. But more often the distribution will contain a handful of possibilities. Our job is to play in such a way as to maximize our EV not against the single most-likely hand, but against the range of plausible hands an opponent could hold.

And in order to maximize our EV in situations involving multiple opponents with hand distributions, we have to know how to calculate our EV in situations involving multiple opponents with hand distributions, using available tools such as PokerStove and, of course, by writing code in our language of choice. Stay tuned.