Hi, complete beginner here. Hope I'm posting this all correctly and in the right place.

In a recent hand (online, No-Limit Hold'em) I was dealt AA, the flop was (something like) 39A, the turn was K, and the river was K. At this point it was only myself and one opponent, and the opponent made a huge raise (approx 10x pot size), which I called, and I want to see if I am thinking about the math of this correctly.

Before I breakdown the thinking I did afterwords, I'll say that my gut instinct in the moment (based on the bet size) was that their hole cards were AK, and that they simply thought their KKKAA full house was the nuts, and were failing to consider anything else at all, including my AAAKK full house. This turned out to be true, but it was simply a snap call on my part, and I was frankly nervous to make it. But like I said, my questions here are about the math. Afterwords, I wanted to know what my actual chances of winning this hand were. My first attempt at considering the math of this was that I broke it down into "categories" of hands that my opponent might have. I came up with 4 categories of hands.

1) they have KK, in which case I lose.
2) they have AK, in which case I win.
3) they have KX, in which case I win.
4) they are bluffing, in which case I win.

After thinking about it in these terms for awhile, I realized that there is an obvious issue with this, and that issue is that my "categories" are essentially arbitrary. In the above breakdown, for example, I would see myself as having a 75% chance to win. But what if I chose to break it down differently? What if I decided that the categories were: 1) They have KK, therefore I lose, or 2) they have literally any other cards than that, and therefore I win. Now, suddenly, by my "categories" I have a 50% chance to win. So, obviously this type of thinking doesn't get me to a mathematically correct answer of my winning chances. But yet, while my categories are essentially arbitrary, they aren't literally random, because they are based on what I believe might be likely, so they must hold some kind of weight in terms of decision making, right? (I suppose this is where the concept of "ranges" might start to come into play?)

Realizing I was wrong there, I then thought that the way to determine the answer is to simply say that the only cards that beat me are KK, and work out the math based on that. Is this the correct way to look at it? Do I have a 99.55% chance of winning the hand? (Chances of being dealt any specific pair = .45% ; 100 - .45 = 99.55) Or are there more factors present in the situation than just that?

As a further question: if my opponent makes a raise that is so large as to make what's in the pot already seem insignificant, am I correct in basically seeing that as 2:1 pot odds? That was the case here, the opponent made a raise that was greater than 10x the existing pot. When that happens, should I simply think of it as (essentially, but slightly worse than) 2:1?
[Edit: I meant to say 1:1 pot odds, i.e. 50%]

In the end, was calling correct? It certainly seems so.

And finally, if calling was indeed correct, does that mean that going all in would have been more correct?

Thanks for any input on these specific questions, or how I am generally thinking about things as a complete beginner.

If you have aces full on the river and someone else is playing back at you, you get it in, if the other guy has quads it's just a cooler

that's true, but the question was about the heuristics, not the specific hand

the way you should approach this spot (and basically every other spot) is by counting possible combos of cards. There are 2 kings and 1 ace left after discounting the cards you know. This means that there is 1 combo of quads possible (both kings) and 2 combos of AK (the ace with first king or the ace with the other king).

Bluffs in such spots are very rare, play like that with something weaker than ak would be a massive overplay, so you can simplify this spot into you facing 3 possible combos in your opponents range. You beat to of them and lose to one, so you should expect to win around 66.6% of time. To call a 10x pot bet, you need to be good roughly 48% of time, which makes this a very easy call - just don't be too surprised when you lose to quads

oh, and one more thing: while this is a good learning spot (there aren't many combos, so it's easy to simplify), don't focus on hands like this one when trying to learn. They are extremely rare and have very little impact on your overall results. What really matters are pots that are small, but common, for example you raising a medium-strength hand, getting one caller and flopping a middle pair.

Thank you for the responses.

From what I'm reading here: conceptually it's about ranges, mathematically it's about combinations.

I will look into both of these subjects. Thanks!

Conceptually your mistake is assuming that all outcomes that you can postulate for a given situation are equally likely. That is generally not true. The only time it is true is when you have some symmetry in the situation that leads you to believe it is. For example, what is the probability that the first card you look at in a hand is the ace of spades? In that situation we have no reason to believe that the ace of spades is either more or less likely to be our first card than any other option. Therefore we ascribe the probability of 1/52 to this event.

Note that we can’t solve this problem by saying “our first card either will be the ace of spades or it will not. Therefore the probability is 1/2”. The probability of the two events “ace of spades” and “not ace of spades” are NOT equal. There are a lot more ways that “not ace of spades” can occur, so that event is much more probable (51/52 to be exact).

In your example KK has fewer ways to occur (1 combination) than “not KK” does, regardless of what other hands we think our opponent might have. It’s not valid to say he has KK 1/2 of the time because of this.

I'm curious, OP, how did you get to the river with two people having monster hands, and there being so little in the pot that he could bet 10X and still have money left over? Remembering the action in the hand can also help you make decisions - for instance most of the time when players have AA and KK it will get all in preflop. Then you don't have to worry about a massive bet on the river. Since you didn't, either 1) he doesn't have KK; or 2) you never bet/raised because you wanted to trap. So there might be only 1 combo of KK, but given the preflop action (which you should know, but we don't) the probability of that one combo is very low. Also, if he has quads, and the pot is very small, and he knows you don't have a K, would he bet so much when you will probably fold 99% of the time to such a large bet.

I have 2 points here: 1) the play of the hand is as much a part of your decision making process as just the combos/math portion of it; and 2) because of that first one, you should get in the habit of remembering the play of the hand, including the positions of players, whenever you post a hand. I know that your question wasn't really about that hand - but anytime you are posting about a situation (even a hypothetical one) it is best to actually clarify the entire situation.

Relevant information:

Number of players dealt in the hand; positions of the players who are directly involved in the hand (don't worry about all the people who fold); the stack sizes, the action on all streets; and any reads you have on the villain.

As for this hand - as others have said, don't worry about it. You want to get it all-in, and if he turns out to have the one combo that beats you - well that's poker. If you have the top full house and someone else has second full house and you don't get it all in by the river you are missing out on a lot of value.

I postulate a 1/8 chance that the first card is an ace of spades. After all, it's an ace or it isn't, and then there's clearly a one in four chance the ace will be a spade!

Yeah, that is the conclusion I was sort of figuring out in my first way of looking at it. I realized that I was making somewhat arbitrary distinctions about the possibilities. What I wasn't thinking about was yet was ranges. I wasn't picturing how my opponent would have gotten into this spot with me, all I knew is that they were there. Well, I guess the only "range" I had them on was something like "Gee, they must have a real strong hand to be here."

And I don't know how to make a second quote, but to VBAces:

Yes, that is something I've picked up on over the last couple weeks of watching poker vlogs. Every discussion of a hand focuses on the positions. My one and only poker playing friend stressed that to me when I told him I had just started playing. He said to be thinking about position from the very start. So, I've been trying to concentrate on that.

Thanks again for all the replies.