Author's Note
This article will be published in my book "The Theory of Poker". So, the target audience for this article is less advanced players.
I've said many times, "It's not a good strategy to "assume that the opponent has a hand of cards" and then continue playing cards based on that speculation. You should speculate that he has a variety of cards, and then weight the average of those hands to come up with the best way to play overall. While the book would have been more general, I'll be more specific about Texas Hold'em, especially No Limit Texas Hold'em.
If you count the suits, there are a total of 1,326 starting cards in Texas Hold'em. The first card is one of the 52 cards, and the second card is one of the 51 cards. You multiply these numbers by dividing by 2 because the order of the two cards is uncertain.
Cards like AK have 16 different combinations, 4 x 4, 12 cards with different flowers in AK, and 4 cards with different flowers in AK. The same is true for 95 or any other unpaired card. But there are fewer pairs of combinations. Each pair has 6 different combinations, and you count. This means that if someone always makes a big raise with AA, KK or AK instead of other cards, there is a good chance of beating him if you get a smaller pair (16:12).
However, you don't have an advantage in holding this pair. If you do it all, you can win 9 times out of 16 times he has AKs. However, only 2 times out of 12 he has a big pair, that is, you can win 11 times out of 28. Obviously, you can use similar techniques in many other situations to figure out your full chances.
Sometimes you have to adjust the number of your opponent's hand combinations because you have one or more of these cards in your own hand or on the deck. If you get an A, or an A on the deck, his 6 combinations with AA are reduced to 3. If you see two A's, the combination is reduced to 1. Unpaired cards are a little more complicated. Using AK as an example, seeing this card can reduce the number of combinations from 16 to 12, 9, 8, 6, 4, 3, 2, or 1, depending on how many cards you see.
An important reason to know how to combine numbers is that if you roll all the way down before the flop, you can estimate your chances. Let's take a simple example, you get a pair, and the opponent is AA, KK or AK. In this case, you need 8:5 pot odds to be able to call.
Another reason is whether it is profitable to estimate the total down-fold. For example, suppose there is only one big blind. The blind is $1, your last move, and the chips are $10. Your hand is two 2s and you're thinking about them all. Suppose you think that any two cards greater than 9 or pairs greater than 22 will be called. First, note that the total possible combination of opponents is reduced to 50 x 49/2, which is 1,225. He would call with 12 x 6, or 72 combined pairs, plus 16 x 10, which is 160, which is a non-pair of 160. Since you get a pot odds of 10:1, if you always lose when you are called, then your play is very unprofitable.
However, this is not the case here. You can win out of 80 times out of 160 times he didn't have a pair and 72 times he had a pair of 12 times. You win $10 92 times, you lose $10 on 140, and the total is $480. On top of that, you win $1 every time he folds before flop. The number of times this happened was 1,225 - 232, which is 993 times. That $993 brings your net profit to $513, or about 40 cents per hand. So, obviously, it's better to do it all than to fold. That doesn't mean it's the right way to play, because sometimes a flat heel or smaller raise will lead to a higher EV. But this should not be the case.
Another pre-flop situation requires you to be familiar with the opponent's range and all the combinations. That's when you think about doing "loose 3bets". In limited-stake Texas Hold'em, 3bet doesn't let the raiser fold. But in the no-limit bet, 3bet usually does. Players who are fairly loose in the blind position need good cards to call or 4bet, so even if the 3bet player can't see the flop (because the opponent usually folds or counter-adds to make him fold), this play is profitable.
For example, suppose an opponent places a big blind (the first into the pool) at the back of the position to triple, and you think he will do so with any pair of about 22 or any card above 9. However, if you reverse the big blind to 10x, he can only call with 99 or above or AK and AQ. If you don't play blinds, you're risking 10 to win 4.5. So, you have to have a 70% success rate. He has 232 hands of jackpots, but will only use 68 of them to call you back (with a few adjustments based on the cards in your hand). So, this style of play seems to be a bit profitable.
But that may not be the case, as someone in the blinds may get the big card. And then again, this calculation assumes that you don't just get called and then win. Since this is sometimes the case in real life, 3bet is worth it when these conditions are met. Remember, though, that just because one style of play is profitable doesn't mean it should be played that way. This just means that it is stronger than the fold. Flat heels or different numbers of raises may be better.
Another time to use various ways of thinking is in the flop circle. Suppose someone places your big blind spot in the front position, and you use JT to call him a heads-up. You think his range is 88+ cards, plus AK, AQ, AJ Flush and KQ Flush. The flop is the K-10-2 Rainbow card. You pass the cards, he plays them all. You think he's going to play this way with the whole range, and your chances of winning are as follows:
He has 6 88 combinations, and you can win 5.5 times at this time.
He has 6 combinations of 99s and you can win 5.5 times at this point.
He has a combination of 1 TT and you can hardly win.
He has a combination of 3 JJs and you can win about 0.3 times.
He has 6 QQ combinations and you can win about 1.2 times.
He has a combination of 3 KK and you can hardly win.
He has 6 AA combinations and you can win about 1.2 times.
He has 12 AK combinations and you can win about 2.5 times.
He has 16 AQ combinations and you can win about 10.5 times.
He has 3 AJ flush combinations and you can win about 2 times.
He has 3 KQ flush combinations and you can win about 0.5 times.
(Remind readers who don't know how these numbers come about, this is mostly based on the number of fills, and I suggest you look at any book that explains the basic Texas Hold'em probability.) ）
In this example, we conclude that your chances of winning are about 29.2 out of 65. This number will be higher than the number you get based on his range. Unless he makes a lot of money, you can easily call.
Apparently, expert-level Texas Hold'em players will observe more details. However, these are all technically the same. If you're worried that this will require mathematical talent, then don't worry. I assure you that most of these Texas Hold'em experts are not mathematical geniuses either.