I was hoping thus forum wouldn't mind helping me with this scenario that came to my mind:

Would you rather be all-in preflop holding

JsJh vs AcKd at a LIVE table while:

A.) 9-handed table

B.) Heads-Up

Note: This is basically mental gymnastics while bed ridden for a couple of days from having wisdom teeth removed and being on light pain meds. 🥺🤕🥴

I'm sitting here pondering mathematical models & real world situations.🤷


Just to be clear:

There is no preflop action aside from player-A who open jams AKo and "accidentally shows his hand."

You are in the BB with JJ.

Same goes for heads-up and as a 9-handed scenario.

This is basically a card removal question.

We always assume that the "dead cards" are unknowable, which is fair, BUT in this case the FOLDED CARDS carry equity implications that we CAN draw information about.

__________

JsJh: ~57.3%

AcKd: ~42.7%

Note: While we can't know the exact cards folded beyond deductive reasoning about opening ranges by position, player tendencies, etc. We DO KNOW that X-number of cards have been randomly removed from the deck. Card removal hurts AcKd more than JsJc at a 3-1 ratio.

Heads-up:

There are 3 "burned" cards from the deck (dealer).

9-handed:

There are 14 cards removed from the deck from the 7 folded hands plus 3 "burned" cards (dealer).

Preflop:

9-HANDED:
From these 17 cards (HU) there is a 3-1 ratio that an Ace or King (6) has been removed vs a Jack (2).

Heads-up 3-cards removed

vs.

9-handed 17-cards removed

_______

Card removal impact:

2 Jacks: (-4.2% average loss per card)

57.3% -> 53.2% -> 48.9%

Loss of value per card JsJh

-4.1%, -4.3%


6 Aces & Kings: (-6.3% average loss per card)

42.7% -> 38.2% -> 33.1% -> 27.6% -> 20.8% -> 13.3% -> 4.9%

Loss of value per card AcKd

-4.5%, 5.1%, 5.5%, 6.8%, 7.5%, 8.4%

____________

I guess one could argue that each player's fold provides additional information about the folded cards, but...

The range deduction ideas open up a can of worms that is extremely complex even with a HUD and a large sample imo. (At least for me and possibly the purpose of this discussion)

_____________
Let me try to explain it an additional way:

There are 3-Aces & 3-Kings (6-cards that hurt AcKd equity).

There are 2-Jacks (2-cards that hurt JsJh equity)

Hence 3-1 ratio of equity loss for each hand.

For each card that is removed from the deck there is a percentage, 3-1 ratio, that AcKd equity is hurt more than JsJh.

Take the remaining 48 cards and remove 24 of them. Which hand is impacted more by this?

Remove 43 cards, etc.

Like I stated above this is mental gymnastics, possibly a waste of time and definitely something that came to mind, bc of my current state of being. (Possibly apophenia) BUT, I do believe that thinking about poker from different angles is important. I don't expect a breakthrough regarding tactics or strategy, but I do think that I'd prefer JsJh vs AcKd with 7-random hands dealt & folded at a 9-handed table vs heads-up for a prop bet. 🤕🥴🤪

_________

ADDITIONALLY: (For the argument that all cards removed from the deck are random and therefore have no bearing on either hand's "equity.")

What are the odds of pulling an Ace (3), King (3) or Jack (2) from a deck of 48 cards excluding the:

JsJh &

AcKd

23:1 is 4.17% chance of the next card pulled being one of the two remaining JACKS.

42:6 is 12.5% chance if the next card pulled being one of the remaining 6 ACES or KINGS.

____

So my point is when we say don't know what the chances of pulling a particular card from the deck & adding it to the discard pile are & that it's all random and blank is incorrect, right?

We CAN draw slight conclusions about any random card being removed from the deck.

Also, is it appropriate to think of FOLDED CARDS as cards that have been processed through a FILTER?

And burned cards as random removal?

Since we DO KNOW that removing the 3-Aces, 3-Kings and 2-Jacks from the deck impact percentages.

What's stopping us from running the impact on the "run out" for each remaining card?

Since there are more cards that hurt the AK from the remaining 48 -> can we then deduce that each card removed from the deck is more detrimental to the AK, which is currently behind?

If there were no more cards in the deck AK has lost.

48-remain=AK has a 42.7% chance.

0-remain=AK loses

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There is more that I would love to ask you, but I'll start here if there's interest.

Thank you in advance for any info that you can give me.
This question is haunting me.

The equity is only affected by the players' folding ranges. With the JJ I'd slightly prefer a HU table. If 9max and 7 players folded, A's and K's aren't quite as removed as the other cards because AA/KK wouldn't have been folded.

The equity isn't affected by completely unknown mucked/burned cards. The problem with your analysis is that you didn't take a weighted average.

Consider, for instance, the chance of the 1st flop card being an A or K. Suppose 7 players folded without even looking at their cards.

P(A/K) = P(0 A/K removed)*P(A/K given 0 removed) + P(1 removed)*P(A/K given 1 removed) + ...

If you crunch all of that, it will come out to 6/48, the same as if you simply ignored all the unknown cards.

We can see the same thing in the HU example, where only the burn card is to be considered:

P(A/K) = (42/48)(6/47) + (6/48)(5/47) = 6/48, same as ignoring the burn card.

You see, there are two opposing effects that cancel out: an A/K can be removed and shrink the probability by a good chunk, but this happens less often than a non-A/K being removed (the majority of the time) and slightly increasing the chance of the next card being A/K.

The same fact can also be deduced without math: splitting a deck into a larger and smaller pile doesn't magically affect any card's probability of being an A or K. The top card and bottom card are both 6/48. The 5th card in Pile #1 is 6/48 and the 14th card in Pile #2 is 6/48. Doesn't matter how many piles you make nor their sizes.