A weird odds situation - a case for drawing to big overcards

[James282]

I am not sure as to the utility of this - but it intrigues me a little. When calculating odds on later streets in hands, we typically this basic formula:

Let's say I have AK. I raise in the CO and only the BB calls. The flop is a rainbow 458. The BB check-raises me and at this point I know he must have a pair(to simplify the calculation). The pot is kinda small, so I peel the turn. The turn is a seemingly innocuous queen. He bets again and since I think he won't defend Q8 or worse I know I will often have 6 live outs here. So I figure, 6 of the 46 cards left are an ace or a king that will likely improve me to the best hand.

But this seems to omit much of the information that we have. The hand is heads up - which means that 1 of the cards in all of my opponents was not an ace or a king - assuming nobody folds AK, AA, or KK in situations like this(which is a pretty safe assumption, especially at higher limits). So aren't my odds more like 6/37? Aren't there many more situations we could conceivably think about this way?
-James

[tongni]

I didn't understand, but what about reverse domination?

[James282]

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I didn't understand, but what about reverse domination?

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I assume that people account for that sort of thing already. This is an assumption that is independent of reverse domination, if you are drawing dead, etc. which most people should already be accounting for.
-James

[Justin A]

I'm not sure if I'm reading your post correctly. This doesn't seem to account for all the Ax and Kx hands that will be thrown away.

[James282]

It does. There is one unknown card in each person's hand(it could be an ace or a king) - we know that the other card was not an ace or a king.
-James

[CardSharpCook]

it is a little complicated but the net effect is going to be minimal. Of the what 1300 possible hand combos? 6 are AA, 6 are KK, and 18 are AK. So that is 30 hand combos (or 2.4%) that we know don't exist. However, that no one has a plurality of aces and kings affects the odds that people hold ugly aces and kings. How much so? Geez, I don't know. My guess would be that knowing that AA/KK/AK don't exist, we can increase the % chance of seeing an ace or king fall on the turn by a guesstimated .3%. It's really gonna be minimal man.

[James282]

Okay, that may be true that the effect is minimal. I am not a math guy. I don't see why we can't discount a full 7 cards though - we know for a fact that 1 of the cards folded was not an ace or a king, so why can't we just take that into account? I'm really asking here.
-James

EDIT: Also, it's not that we know AA/KK/AK don't exist, it's that we know for sure that exactly 1 card in each folded hand was not an ace or a king. We don't need to spend any time speculating what the other card was as it is a true "unknown."

[CardSharpCook]

again, it goes to hand combinations, there are 18*13+6 or 222 hands with at least one ace in it. I think we can *know* that AA,AK,AQ,AJ do not exist, so 60 of the combos are not there, we are left with 162 combos that include an ace. For kings, we *know* that AK,KK,KQ do not exist so 42 do not exist, 180 do. If *knowing* that these hands did not exist did not effect the odds of hands like Ax and Kx existing, then the chance that the next card is an ace is increased from 6.7% to 9.1% and that the next card is a king from 6.7% to 8.2%. However, knowing that AK,KK,AQ,AJ,AA,KQ do not exist increases the likelyhood of Ax/Kx. I don't really know how to judge this though. You can't simply decrease the total number of hand combes by 102 (the # of combos we *know* don't exist). That these hands don't exist in effect increases the number of aces/kings in the deck making it more likely that a random player will be dealt an A/K. Also, you are ignoring other things that we also *know* i.e. that 77-QQ don't exist not to mention a number of hands that have weighted levels of existence (QJs may or may not have been folded in EP, but probably didn't exist in MP where it would be bet). Follow?

Too many variables and too hard to keep track of them all. It has very little effect on the deck though.

[Justin A]

[ QUOTE ]
Also, it's not that we know AA/KK/AK don't exist, it's that we know for sure that exactly 1 card in each folded hand was not an ace or a king. We don't need to spend any time speculating what the other card was as it is a true "unknown."

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The error in your logic is that by saying that at least one of the cards is not an ace or king, the other card is no longer an unknown. It is in fact now more likely that the other card is an ace or king. I did some quick calculations that aren't complete but are probably close enough, and we can probably discount about one card from our unknowns when we have seven people that folded. That's just for the assumption that they don't fold AA KK and AK. Add in other hands such as AJ AQ and KQ, you can discount slightly more.

[James282]

I talked to Justin A about it and he basically agrees. His calculations led him to believe that it maybe makes a 1 card difference. Maybe I am just a freak that is persuaded by lame clumping theories [img]/images/graemlins/smile.gif[/img]
-James