EOR and EV of each card in dealer’s hand for UTH?

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May 22nd, 2026 at 7:25:09 AM permalink

Any starting 2-card hand in UTH for which you bet 4X is profitable for player.

Let's count the number of times that a rank appears in a (+EV) 4X opening hand. Unsuited hands, including pairs, will count as 1 and suited hands will count as 0.25.

Ace___13
King___10.75
Queen___7.5
Jack___5
Ten___5
Nine___4
Eight___4_____A8, K8, Q8, 88
Seven__3.25
Six___3.25
Five___3
Four___2.25
Three___2.25_____A2, K2s, 33
Deuce___1.25_____A2, K2s

When eliminating ranks, this is a relative frequency of profitable starting hands in UTH that are eliminated. Of course, AA is far more profitable than 33 so this is far from a complete story.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

aceside

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May 22nd, 2026 at 3:56:20 PM permalink

A little revision here. For every pair, you count once; however, for every none-pair, you should count twice. For example, the 8-containing hands have these possibilities: A8, 8A, K8, 8K, Q8, 8Q, 88. So, there are seven counts of appearing.

There are also suited 8-containing hands: J8, 8J.

gordonm888
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May 22nd, 2026 at 7:05:57 PM permalink

Quote: aceside
A little revision here. For every pair, you count once; however, for every none-pair, you should count twice. For example, the 8-containing hands have these possibilities: A8, 8A, K8, 8K, Q8, 8Q, 88. So, there are seven counts of appearing.

There are also suited 8-containing hands: J8, 8J.
link to original post

My nomenclature was bad: when I referred to an unsuited starting hand, I really meant {both unsuited and suited}.

As for pairs having fewer combinations than unpaired hands -yes, but pairs tend to have a larger +EV than unpaired hands. And all ranks except deuces had a pair in their frequency count, so it didn't seem like a big deal. I was just trying to illustrate the principal that EV vs rank removed should have a choppy declining trend because the number of profitable hands varies non-smoothly with rank. I was too lazy to do actual math.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

Mental

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May 30th, 2026 at 7:53:09 PM permalink

I had a calculation running when I left for the Indy 500, but I inadvertently ran the wrong calculation. I am 9/13ths of the way through the right calculation, but it won't finish before I leave tomorrow for a 7-day bike trip. I am posting the partial results, and I will update with full results when I can.

Red bars are computer perfect optimal strategy EVs for when a dealer is know to have a certain card. Blue bars are the EV when you play BS when the dealers hand contains a certain card. The ranks left-to-right are 23456789TJQKA.

If the dealer hand is known to contain a nine of clubs, your EV is +0.08 if you play optimally. Your EV is -0.03 if you play BS.

Playing BS when you know the dealer holds an ace of clubs is very bad. You will often be raising 4x with hands like TJo or 9Ko that are dominated. You will seldom win when the dealer qualifies.

Gambling is a math contest where the score is tracked in dollars. Try not to get a negative score.

aceside

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May 31st, 2026 at 6:11:59 AM permalink

This is a lot of calculation, but probably not much usefulness to players. Let’s be straightforward. If one dealer card is a known Ace, should player 4x raise his AJ offsuit hand? And 10-10 pair hand?

Mental

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May 31st, 2026 at 7:17:07 AM permalink

Quote: aceside
This is a lot of calculation, but probably not much usefulness to players. Let’s be straightforward. If one dealer card is a known Ace, should player 4x raise his AJ offsuit hand? And 10-10 pair hand?
link to original post

It is a slow calculation because I recalculate the basic strategy for every hand and then do the quicker calculation of the EV for the hand that is constrained to have a particular card in the dealer hand. I could speed this up a lot if it mattered to me. I cannot run these calculations on battery power for very long -- an hour at most.Will try to finish it in the hotel tonight.

I include a snippet of the results files for high cards against ace (suited or not suited to the players hole cards). ALL means normal game, DLR means the particular card is in the dealer hand. Note that the presence of an ace improves the player's EV for some hands. You would be shocked to find that some very weak hands are not harmed by having a dealer ace. EV columns are Raise, Check, Best EV.

9,10,1,1.024156,0.693598,1.024156,Jc,Qc,Jc_Qc,Ac,ALL

9,10,1,-1.332715,-0.593780,-1.332715,Jc,Qc,Jc_Qc,Ac,DLR

9,10,3,1.024156,0.693598,1.024156,Jc,Qc,Jc_Qc,Ad,ALL

9,10,3,-0.884039,-0.236196,-0.884039,Jc,Qc,Jc_Qc,Ad,DLR

9,11,1,1.227500,0.767587,1.227500,Jc,Kc,Jc_Kc,Ac,ALL

9,11,1,-1.477953,-0.761039,-1.477953,Jc,Kc,Jc_Kc,Ac,DLR

9,11,3,1.227500,0.767587,1.227500,Jc,Kc,Jc_Kc,Ad,ALL

9,11,3,-1.020862,-0.395739,-1.020862,Jc,Kc,Jc_Kc,Ad,DLR

9,12,3,1.489507,0.886020,1.489507,Jc,Ac,Jc_Ac,Ad,ALL

9,12,3,1.155341,0.646318,1.155341,Jc,Ac,Jc_Ac,Ad,DLR

9,22,3,2.749739,1.650960,2.749739,Jc,Jd,Jc_Jd,Ac,ALL

9,22,3,1.330751,0.693211,1.330751,Jc,Jd,Jc_Jd,Ac,DLR

9,22,3,2.749739,1.650960,2.749739,Jc,Jd,Jc_Jd,Ah,ALL

9,22,3,1.365341,0.723279,1.365341,Jc,Jd,Jc_Jd,Ah,DLR

9,23,3,0.455681,0.224297,0.455681,Jc,Qd,Jc_Qd,Ac,ALL

9,23,3,-1.668766,-0.765439,-1.668766,Jc,Qd,Jc_Qd,Ac,DLR

9,23,3,0.455681,0.224297,0.455681,Jc,Qd,Jc_Qd,Ad,ALL

9,23,3,-1.670806,-0.766850,-1.670806,Jc,Qd,Jc_Qd,Ad,DLR

9,23,6,0.455681,0.224297,0.455681,Jc,Qd,Jc_Qd,Ah,ALL

9,23,6,-1.623147,-0.736860,-1.623147,Jc,Qd,Jc_Qd,Ah,DLR

9,24,3,0.691991,0.318649,0.691991,Jc,Kd,Jc_Kd,Ac,ALL

9,24,3,-1.794351,-0.926809,-1.794351,Jc,Kd,Jc_Kd,Ac,DLR

9,24,3,0.691991,0.318649,0.691991,Jc,Kd,Jc_Kd,Ad,ALL

9,24,3,-1.798003,-0.928510,-1.798003,Jc,Kd,Jc_Kd,Ad,DLR

9,24,6,0.691991,0.318649,0.691991,Jc,Kd,Jc_Kd,Ah,ALL

9,24,6,-1.747178,-0.900762,-1.747178,Jc,Kd,Jc_Kd,Ah,DLR

9,25,3,0.991767,0.461261,0.991767,Jc,Ad,Jc_Ad,Ac,ALL

9,25,3,0.609950,0.196573,0.609950,Jc,Ad,Jc_Ad,Ac,DLR

9,25,6,0.991767,0.461261,0.991767,Jc,Ad,Jc_Ad,Ah,ALL

9,25,6,0.632165,0.231214,0.632165,Jc,Ad,Jc_Ad,Ah,DLR

10,11,1,1.312620,0.816408,1.312620,Qc,Kc,Qc_Kc,Ac,ALL

10,11,1,-1.477453,-0.775550,-1.477453,Qc,Kc,Qc_Kc,Ac,DLR

10,11,3,1.312620,0.816408,1.312620,Qc,Kc,Qc_Kc,Ad,ALL

10,11,3,-1.019966,-0.407208,-1.019966,Qc,Kc,Qc_Kc,Ad,DLR

10,12,3,1.572705,0.935691,1.572705,Qc,Ac,Qc_Ac,Ad,ALL

10,12,3,1.578266,0.872101,1.578266,Qc,Ac,Qc_Ac,Ad,DLR

10,23,3,3.018649,1.821642,3.018649,Qc,Qd,Qc_Qd,Ac,ALL

10,23,3,1.449309,0.762683,1.449309,Qc,Qd,Qc_Qd,Ac,DLR

10,23,3,3.018649,1.821642,3.018649,Qc,Qd,Qc_Qd,Ah,ALL

10,23,3,1.484579,0.793271,1.484579,Qc,Qd,Qc_Qd,Ah,DLR

10,24,3,0.782942,0.370490,0.782942,Qc,Kd,Qc_Kd,Ac,ALL

10,24,3,-1.795186,-0.941404,-1.795186,Qc,Kd,Qc_Kd,Ac,DLR

10,24,3,0.782942,0.370490,0.782942,Qc,Kd,Qc_Kd,Ad,ALL

10,24,3,-1.797227,-0.942414,-1.797227,Qc,Kd,Qc_Kd,Ad,DLR

10,24,6,0.782942,0.370490,0.782942,Qc,Kd,Qc_Kd,Ah,ALL

10,24,6,-1.746476,-0.912429,-1.746476,Qc,Kd,Qc_Kd,Ah,DLR

10,25,3,1.080609,0.514408,1.080609,Qc,Ad,Qc_Ad,Ac,ALL

10,25,3,1.062666,0.437359,1.062666,Qc,Ad,Qc_Ad,Ac,DLR

10,25,6,1.080609,0.514408,1.080609,Qc,Ad,Qc_Ad,Ah,ALL

10,25,6,1.081814,0.470766,1.081814,Qc,Ad,Qc_Ad,Ah,DLR

11,12,3,1.658326,0.986740,1.658326,Kc,Ac,Kc_Ac,Ad,ALL

11,12,3,1.989133,1.094723,1.989133,Kc,Ac,Kc_Ac,Ad,DLR

11,24,3,3.289359,1.993532,3.289359,Kc,Kd,Kc_Kd,Ac,ALL

11,24,3,1.564299,0.830017,1.564299,Kc,Kd,Kc_Kd,Ac,DLR

11,24,3,3.289359,1.993532,3.289359,Kc,Kd,Kc_Kd,Ah,ALL

11,24,3,1.600228,0.861114,1.600228,Kc,Kd,Kc_Kd,Ah,DLR

11,25,3,1.171914,0.568928,1.171914,Kc,Ad,Kc_Ad,Ac,ALL

11,25,3,1.501538,0.675084,1.501538,Kc,Ad,Kc_Ad,Ac,DLR

11,25,6,1.171914,0.568928,1.171914,Kc,Ad,Kc_Ad,Ah,ALL

11,25,6,1.517642,0.706500,1.517642,Kc,Ad,Kc_Ad,Ah,DLR

12,25,3,3.601073,2.192924,3.601073,Ac,Ad,Ac_Ad,Ah,ALL

12,25,3,3.991335,2.221799,3.991335,Ac,Ad,Ac_Ad,Ah,DLR

Gambling is a math contest where the score is tracked in dollars. Try not to get a negative score.

gordonm888
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May 31st, 2026 at 8:18:12 AM permalink

Quote: Mental
You would be shocked to find that some very weak hands are not harmed by having a dealer ace.
link to original post

No, that's poker.

In conventional poker you can open with, say, a 76s. When the flop comes if it is 2 or 3 high cards you will be folding (unless they're in your suit); what you are hoping for is a flop with 2 or ideally 3 low cards that pair you or give you straight possibilities. If you know your opponent has an Ace then that is no different than when he has a Jack -they are both 'overcards', higher than your 7 or 6. You assume he has high cards like T,J,Q,K,A and you are only continuing only if the flop is low cards or gives you a flush draw.

In UTH, it is the same principle. If you open with 54, you are checking and you should expect that dealer has higher cards than yours. Doesn't matter whether dealer has an AK as opposed to a 98 - they're all higher than your two crappy cards. You are hoping that dealer doesn't pair and you get one pair - or that you make a straight or two pair.

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

aceside

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May 31st, 2026 at 8:10:12 PM permalink

Quote: Mental
Playing BS when you know the dealer holds an ace of clubs is very bad. You will often be raising 4x with hands like TJo or 9Ko that are dominated. You will seldom win when the dealer qualifies.
link to original post

I’ve been thinking about this part. Knowing a dealer Ace is still very different from knowing a dealer Ace of clubs. So, here is my question. Is your basic strategy the same for these two different situations?

Mental

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June 1st, 2026 at 3:51:04 AM permalink

The calculation finished, but the average of the EORs is off by 0.3%. The average EV for placing all 13 ranks in the dealers hole cards should be the same as the EV of the base game (assuming I use the same BS). I am not sure what to make of the small discrepancy.
0 0.473939
1 0.389140
2 0.303292
3 0.218642
4 0.162909
5 0.104732
6 0.042896
7 -0.026492
8 -0.120932
9 -0.191599
10 -0.310372
11 -0.518471
12 -0.847202

Average -0.024578

I will be riding on the Erie Canalway bike trail until Sunday, and I won't be spending any time on this problem until I get home.

Gambling is a math contest where the score is tracked in dollars. Try not to get a negative score.

gordonm888
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June 1st, 2026 at 5:39:13 AM permalink

Quote: aceside
This is a lot of calculation, but probably not much usefulness to players. Let’s be straightforward. If one dealer card is a known Ace, should player 4x raise his AJ offsuit hand? And 10-10 pair hand?
link to original post

"Poker is a game of high cards"

The fact that an observed Ace in the dealer's hand is a certain suit is immaterial unless both your cards are that suit or unless the board comes out with 3 or more of that suit. Which are infrequent circumstances.

High cards make the dealers hand strong. Low cards make it weak. Why? Read above. "Poker is a game of high cards"

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

aceside

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June 1st, 2026 at 5:53:19 AM permalink

I just looked into the list of hands you calculated. I see you have calculated this (player two cards, dealer exposed)
Jc_Qd,Ad.

But it looks like you haven’t calculated another suit combination, like this one
Jc_Qd,Ah.

By the way, enjoy your bike ride. Are you talking about the 350-mile long route from Buffalo to Albany?

Mental

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June 1st, 2026 at 7:36:23 PM permalink

Quote: aceside
I just looked into the list of hands you calculated. I see you have calculated this (player two cards, dealer exposed)
Jc_Qd,Ad.

But it looks like you haven’t calculated another suit combination, like this one
Jc_Qd,Ah.

By the way, enjoy your bike ride. Are you talking about the 350-mile long route from Buffalo to Albany?
link to original post

Maybe you overlooked it or it is missing from the post you received.
9,23,6,-1.623147,-0.736860,-1.623147,Jc,Qd,Jc_Qd,Ah,DLR

Yes, we left Buffalo this morning and logged 67 miles into a headwind today. We are riding e-bikes, since we we both born in the first Eisenhower administration and we now need a little extra power to make the trip more enjoyable.

Gambling is a math contest where the score is tracked in dollars. Try not to get a negative score.

aceside