My Problem with "# of combinations villain can have

[bigslikk]

There's a certain concept I don't understand. I don't believe in it, but I hear a lot of the seasoned posters around here use it though, so I figure they might be on to something...

The board is QQA

Hero has AK, but, due to preflop / flop action, has a solid read that the Villain has a Queen for trips. Because of this, he checks behind villain, evading his "trap" and hoping for an Ace on the turn. The turn is a third queen for QQAQ. The villain bets in the manner that screams quad queens.

However, the Hero, seeing three queens on the board, figures that the probability of (or the # of combinations that give) the villain quads are now greatly diminished. (Hero might call or something figuring villain for an ace).

Case 2:

Hyper Aggressive 4 bets Hero preflop. Hero calls with kings, figuring that while there are 6 combinations of cards that make AA, there are 12 that make AK, and so it is more likely that the villain holds AK here).

This concept (I suspect I may be presenting it wrong) makes me scratch my head. I am baffled at how this sort of after-the-fact reasoning can be used to estimate how much weight different hands should be given in a range. Isn't there some rule where the probability of an event is not effected by future events?

Analogy: A Bag holds four marbles: two are blue and two are red. Your friend draws a marble without showing it to you. You then draw a bag out of the marble: it is blue. You now figure that your friend probably has a red marble. Isn't this flawed since your friend had a 50/50 shot of having red/blue and has the same likelihood now? Does your discovery of new information after the fact affect the probability of your friends holdings? I say no, it doesn't.

I am sure that I'm mistaken in some manner, I'm just struggling with this concept and would really like to understand it.

[BankItDrew]

Analogy: A Bag holds four marbles: two are blue and two are red. Your friend draws a marble without showing it to you. You then draw a bag out of the marble: it is blue. You now figure that your friend probably has a red marble. Isn't this flawed since your friend had a 50/50 shot of having red/blue and has the same likelihood now? Does your discovery of new information after the fact affect the probability of your friends holdings? I say no, it doesn't.

New information can be applied to past information. Combining the two you now have what I like to call 'super-information...' jk... It's the same sort of idea when we have and and now knowing that villain on the button has a one in 12.5 chance of having an ace. Before we see our cards, this number is different.

btw, you can't draw bags out of marbles.

[allabout]

Think of it like this...take the same bag of marbles. Your friend takes one and you take two. You have two blue marbles so although before you took any marbles you knew there was a 50/50 shot of him having a blue marble, you now know there is a 0% chance of him having a blue marble. Thats why poker is a game of incomplete or partial information.

[cjs55]

Once your friend picks a marble, he either has red or blue. What was once a 50% probability is now a certain answer (that you can guess at)

Certain knowledge can push you towards one answer or another.

When you pick a blue, it means that you were more likely picking from a bag that contained 2 blues and 1 red. Meaning it's more likely your friend began by picking a red.

[Robb]

Case 2:

Hyper Aggressive 4 bets Hero preflop. Hero calls with kings, figuring that while there are 6 combinations of cards that make AA, there are 12 that make AK, and so it is more likely that the villain holds AK here).

This concept (I suspect I may be presenting it wrong) makes me scratch my head. I am baffled at how this sort of after-the-fact reasoning can be used to estimate how much weight different hands should be given in a range. Isn't there some rule where the probability of an event is not effected by future events?

The original probabilities don't change. Our ability to figure them out changes. Think of a coin flip to start a game. The captain sees the coin "heads up" just a split second before the ref's hand covers it up. Of course, the odds were 50-50 of heads vs tails (or near enough not to matter much). But the captain's ability to predict the outcome of a probability-based experiment has improved greatly.

Same deal with Case 2. If we know villain will 4-bet AA and AK (ignoring KK which he should also be betting, if he's willing to bet AK), how many possible hands could he have? BTW, there aren't 12 possible ways to make AK. There are 16 total, 8 of which would include the two kings in our hand, the other half of which involve the other two kings.

So he has 8 AK's possible and 6 AA's. Even though knowing where two kings are dramatically reduced MY ESTIMATE of his original probability of having drawn AK, he still had more ways to make AK than AA given the now-known information.

By the way, there is no rule about probabilities vis-a-vis future events. Probabilities are either known or not, and either change or not, depending upon the experiment. What changes a lot is our ability to predict what happened in the past based on learning information before we have to make our prediction.

This type of inference is called "conditional probability." We can use it to effectively to reason backwards, adding in the new information as we go. As long as we can gather the information before we have to bet, we can revise our probabilities.

[Pelion]

Analogy: A Bag holds two marbles: one is blue and one is red. Your friend draws a marble without showing it to you. You then draw a marble out of the bag: it is blue. You now figure that your friend probably has a red marble. Isn't this flawed since your friend had a 50/50 shot of having red/blue and has the same likelihood now? Does your discovery of new information after the fact affect the probability of your friends holdings? I say no, it doesn't.

[bigred]

Analogy: A Bag holds four marbles: two are blue and two are red. Your friend draws a marble without showing it to you. You then draw a bag out of the marble: it is blue. You now figure that your friend probably has a red marble. Isn't this flawed since your friend had a 50/50 shot of having red/blue and has the same likelihood now? Does your discovery of new information after the fact affect the probability of your friends holdings? I say no, it doesn't.

New information can be applied to past information. Combining the two you now have what I like to call 'super-information...' jk... It's the same sort of idea when we have and and now knowing that villain on the button has a one in 12.5 chance of having an ace. Before we see our cards, this number is different.

btw, you can't draw bags out of marbles.

The bots have discovered super information, we must steal this from them and then kill them.
I think you're taking the probability of what villain has preflop as an absolute. As previously mentioned by many posters, this is only one ingredient to the recipe.

[jyms]

When we are talking ranges your not going to make a bang on decision on what marble he has on the first try. But if you pull a marble out of the bag 1000 times you will know what he has in his hand. Think of it this way, if he pulls a red marble out of the bag, and you pull a marble 1000 times you are going to pull more blue marbles than red, about 2:1. Basing this on information provided, you know that there are only three marbles in the bag, and he has one in his hand. If you were to bet that you could guess his marble, you will be right two out of every three times, which in turn is an EV call for you. Same with cards, if you figure someone to have either AA or AK or even KK, what is on the board and what is in your hand will allow you to base this information as MORE CORRECT over the long term.

[Robb]

The book "The Mathematics of Poker" goes into great depth on conditional probabilities. Don't pick it up unless you're a math geek or John Nash wannabe, tho. Here's their example:

You're playing against an unkown villain who plays 9 of the first 10 hands he's dealt. Which is more likely:

A. He's a 70/10 maniac, or
B. He's a 15/10 TAGG who's hit 9 good hands in a row?

If we assume for the moment he would open the exact same range all the time, his range is a pre-decided probability. But it should be obvious that, given our tiny bit of information, we have to place a much higher probability on choice A than B.

[bigslikk]

When we are talking ranges your not going to make a bang on decision on what marble he has on the first try. But if you pull a marble out of the bag 1000 times you will know what he has in his hand. Think of it this way, if he pulls a red marble out of the bag, and you pull a marble 1000 times you are going to pull more blue marbles than red, about 2:1. Basing this on information provided, you know that there are only three marbles in the bag, and he has one in his hand. If you were to bet that you could guess his marble, you will be right two out of every three times, which in turn is an EV call for you. Same with cards, if you figure someone to have either AA or AK or even KK, what is on the board and what is in your hand will allow you to base this information as MORE CORRECT over the long term.

This makes sense to me. However, how to we reconcile this information with our reads?

Example: Marble bag still has 2 red / 2 blue. Friend, instead of picking one at random, cherry-picks one (you don't see the color). You figure that there's a good chance he drew the red one (say that red is his favorite color, but there's a small-but-decent chance he took blue). You draw a marble (this is your "information"): red. Having drawn the red marble reveals that it is likely (in a vaccuum) that your friend has taken the blue marble. So...?

I think this mimicks the conundrum of the quad queens example I provided in the original post. Aren't reads >> information about probabilities in a vaccum? I mean, when someone four-bets (and you have 22) they technically have the same probability of any hand (except for hands with deuces). However, you'd do well to put the villain on a more limited range (say KK+).

[cjs55]

Yeah, that's no doubt correct. It's gotta be applied to ranges. Any hands he would play in a similar fashion can be assigned ranks of likelyhood with this idea.

For instance, you are playing against a hypothetical nit who only plays strong when he hits a set or better. The board is 447T. You have TT. He raises all in. He either has 77, or 44. If they are both equal probabilities, then your odds of winning the hand are at 50%. But they aren't, because it's much more likely he has 77 than he has 44 due to conditional probability. So your odds of winning the hand are much better than 50%.

[Robb]

Aren't reads >> information about probabilities in a vaccum? I mean, when someone four-bets (and you have 22) they technically have the same probability of any hand (except for hands with deuces). However, you'd do well to put the villain on a more limited range (say KK+).

It's reads + info about probabilities. They compliment each other so we can narrow villain's range. Just because we have a read doesn't mean we can ignore the conditional probabilities involved. They have tendencies, but those tendencies are limited by the cards they had to draw a hand from. If we know later that certain cards weren't available when they drew their hand, we narrow our estimates of what they might have accordingly.

In the marbles example, with villains cherry picking, say we estimate that he'll take red 90% of the time. Then we draw red. Here's how the conditional probability would work:

90%: Villain chooses red (1 red, 2 blue left)
10%: Villain chooses blue (2 red, 1 blue left)

I draw.
For 90% line, I have 1/3 chance to draw red. So .9 * .33 =.3 (approx.)
For 10% line, I have 2/3 chance to draw red. So .1 * .67 =.07 (approx.)

What do I know? I have a 37% chance to draw red, and a 63% chance to draw blue given villain's tendencies. And I drew a red. So I take the 37% (since I know we're dealing with that case), and divide the .3 by it. Then .3 / .37 =.81.

So I figure there's an 81% chance he has red, and a 19% chance he has blue. This is different from the original probabilities because I have new info which improves our estimate of his original action.