3-Card Poker: Percentage of hands dealer qualifies

Hi Everyone,

First time poster with what I'm sure is a simple question for some of you:

What percentage/fraction of the time does the dealer's hand qualify in 3-Card Poker (standard rules with a hand of Queen-high or better qualifying).

Much appreciated!

-RP

Quote: robephil

Hi Everyone,

First time poster with what I'm sure is a simple question for some of you:

What percentage/fraction of the time does the dealer's hand qualify in 3-Card Poker (standard rules with a hand of Queen-high or better qualifying).

Much appreciated!

-RP

There are 22,100 possible combinations in 3 card poker:

Straight flush: 4*12 = 48
Three of a kind: 4*13 = 52
Straight: 12*4^3 - 48 = 720
Flush: 4*13*12*11/6 - 48 = 1096
Pair: 13*6*48 = 3744
Ace high: 64 ranks*60 suit combos = 3840
AKJ
AKT
...
AK2
AQJ
AQT
...
AQ2
...
...
A24

King high: 54 ranks*60 suit combos = 3240
KQT
KQ9
...
KQ2
...
...
K32

Queen high: 44 ranks*60 suit combos = 2640
QJ9
...
QJ2
...
...
Q32

Total qualifying combos: 15,380

Total combos: 52*51*50/6 = 22,100

Dealer qualifies: 15,380/22,100 = 0.6959 = 69.59%

Awesome! Very clear and helpful. Thank you.

Hi and I'm sorry for dredging up this old post. I haven't played 3 card poker yet. I assume that when dealt your cards you can just flip them over and anyone can see them. If that's the case, if there are 3 players in and two of the hands have 5 qualifying (for example) face cards would that affect how you play a less than Q hand? After all, that would mean the dealer has only 7 cards left out of 43 to beat you with a high card. At what point does knowing the qualifying high cards that are out of the deck make betting against dealer qualifying worth it? I know this leaves out a pair or better, but the dealer will mostly qualify with a high card.

The dealer will probably tell you off for showing other players your cards. Some marginal Q's might be worth playing or folding depending on what you see, but remember the dealer can qualify via pairs, straights flushes etc which don't need high cards. In 5-card poker knowledge of As Ks and matching cards did affect your play sometimes, but (I'm guessing) in 3-card poker the effects are not so large and there's quite a percentage advantage for folding to overcome.

Really? I know places where BJ cards are dealt face down, and revealong them was never a problem. As far as the other qualifying hands, they were covered on the thread. Find me the game where the payout is the same and those hands show up anywhere near as much as q to a high and I'm all over that. If I were just betting on the dealer getting pair or better I'd get paid everyday. Dealer qualifies with queen or better high card as the most likely option. Seeing 4 qualifiers may be the equivalent of not having a q, k or a in the deck. I'm just not sure of the math. Don't forget that the odds just need to come down to 49.99 percent and it's worth it

I have a hypothetical question....

If at a full table with 7 players and the other 6 players muck their garbage hands, and you are holding Q64 offsuit (the minimum hand we should play). Should we still play this hand?

Also, could someone please show me the math on the chances of the dealer getting an A, K, or Q only, if all 7 players have garbage hands. I know that its 12/31 (38.7% for just one card), but how do I calculate if for the dealer getting two more cards.

TIA

Quote: Edpokernut

I have a hypothetical question....

If at a full table with 7 players and the other 6 players muck their garbage hands, and you are holding Q64 offsuit (the minimum hand we should play). Should we still play this hand?

Also, could someone please show me the math on the chances of the dealer getting an A, K, or Q only, if all 7 players have garbage hands. I know that its 12/31 (38.7% for just one card), but how do I calculate if for the dealer getting two more cards.

TIA

First Question: If you can assume everyone is playing the proper strategy, then no, you should fold. Too many high cards left. The only card(s) that may have been folded that could help you are other queens in a hand otherwise not high enough to call, which is unlikely. Maybe one other hand like that.

Second Question:

If all seven players have Jacks High, or worse, then that leaves 31 cards as you pointed out. You know that 12 of those cards are Q's, K's or A's (assuming no borderline Queens folded), so here you go:

Actually, I'm not sure if you're wanting the probability of the dealer not getting any of these, or only one of these, but since it's easy...I'll do it for both and we'll call that a bonus.

DEALER NOT GETTING ANY:

(19/31 * 18/30 * 17/29) = 0.21557285873 or 1/0.21557285873 = 1 in 4.63880288962***

***Remember, the dealer can still qualify with pairs, flushes, straights and trips even if that happens. In other words, the dealer will still qualify more than 78.442714127% of the time in this scenario.

DEALER ONLY GETTING ONE HIGH CARD:

(19/31 * 18/30 * 12/29) + (19/31 * 12/30 * 18/29) + (12/31 * 19/30 * 18/29) = 0.45650723025

Simply put, the Dealer will get one and only one high card about 45.650723025% of the time in this scenario.

DEALER GETTING TWO HIGH CARDS: (WHY NOT? WE ALREADY DID ZERO AND ONE)

(19/31 * 12/30 * 11/29) + (12/31 * 19/30 * 11/29) + (12/31 * 11/30 * 19/29) = 0.27897664071

Okay, so the dealer will get two high cards 27.897664071% of the time.

DEALER GETTING THREE HIGH CARDS:

(12/31 * 11/30 * 10/29) = 0.0489432703 or 4.889432703% of the time.

0.21557285873 + 0.45650723025 + 0.27897664071 + 0.0489432703 = .99999999999

Would you have thought, even with all those low cards out, the dealer is more likely to get EXACTLY two Q-A than he is to get zero? I wouldn't have guessed that.

Quote: Edpokernut

I have a hypothetical question....

If at a full table with 7 players and the other 6 players muck their garbage hands, and you are holding Q64 offsuit (the minimum hand we should play). Should we still play this hand?

Also, could someone please show me the math on the chances of the dealer getting an A, K, or Q only, if all 7 players have garbage hands. I know that its 12/31 (38.7% for just one card), but how do I calculate if for the dealer getting two more cards.

TIA

11/34, x 3 as the dealer has 3 cards. Assuming that it's a high card that does it. Remember that you yourself took out the queen.

Quote: Mission146

First Question: If you can assume everyone is playing the proper strategy, then no, you should fold. Too many high cards left. The only card(s) that may have been folded that could help you are other queens in a hand otherwise not high enough to call, which is unlikely. Maybe one other hand like that.

Second Question:

If all seven players have Jacks High, or worse, then that leaves 31 cards as you pointed out. You know that 12 of those cards are Q's, K's or A's (assuming no borderline Queens folded), so here you go:

Actually, I'm not sure if you're wanting the probability of the dealer not getting any of these, or only one of these, but since it's easy...I'll do it for both and we'll call that a bonus.

DEALER NOT GETTING ANY:

(19/31 * 18/30 * 17/29) = 0.21557285873 or 1/0.21557285873 = 1 in 4.63880288962***

***Remember, the dealer can still qualify with pairs, flushes, straights and trips even if that happens. In other words, the dealer will still qualify more than 78.442714127% of the time in this scenario.

DEALER ONLY GETTING ONE HIGH CARD:

(19/31 * 18/30 * 12/29) + (19/31 * 12/30 * 18/29) + (12/31 * 19/30 * 18/29) = 0.45650723025

Simply put, the Dealer will get one and only one high card about 45.650723025% of the time in this scenario.

DEALER GETTING TWO HIGH CARDS: (WHY NOT? WE ALREADY DID ZERO AND ONE)

(19/31 * 12/30 * 11/29) + (12/31 * 19/30 * 11/29) + (12/31 * 11/30 * 19/29) = 0.27897664071

Okay, so the dealer will get two high cards 27.897664071% of the time.

DEALER GETTING THREE HIGH CARDS:

(12/31 * 11/30 * 10/29) = 0.0489432703 or 4.889432703% of the time.

0.21557285873 + 0.45650723025 + 0.27897664071 + 0.0489432703 = .99999999999

Would you have thought, even with all those low cards out, the dealer is more likely to get EXACTLY two Q-A than he is to get zero? I wouldn't have guessed that.

Thank you so much Mission! What got me thinking about this was that I sometimes see players who consistently play their trash hands. As you have shown, the more players that don't have a playable hand, the more of a suicide play it is.

Quote: Edpokernut

Thank you so much Mission! What got me thinking about this was that I sometimes see players who consistently play their trash hands. As you have shown, the more players that don't have a playable hand, the more of a suicide play it is.

You're welcome!

That's absolutely right. That's referred to as, "Composition-Dependent Strategy," and would change some of your more borderline plays, which includes folding the hand in your example. The term, "Composition-Dependent Strategy," is more typically applied to single-deck Blackjack and card-counting (to wit, that is precisely what card-counting is) but it can be applied to other games, as well.

Interestingly enough, if enough players stayed in as a result of having high cards, (low pairs would hurt you as that would mean more high cards for the dealer) then you could be more aggressive in making the Play bet.

Problem is, you don't know what people are staying in or folding on. You might have six others playing 223, 456, 772, Q94, TT8, 369 suited. You'll falsely believe they all have mostly high cards.

Quote: Ibeatyouraces

Problem is, you don't know what people are staying in or folding on. You might have six others playing 223, 456, 772, Q94, TT8, 369 suited. You'll falsely believe they all have mostly high cards.

That's absolutely right, but composition-dependent strategy would rely upon only the cards that you know. I don't know how loosey-goosey the casinos are about letting people look at each other's hands on 3CP as I've essentially never really played it...maybe once. I mean, if someone wanted to make assumptions as to what was being held, they could, but in the strictest sense, that wouldn't be composition-dependent because the fact that someone Plays does not give you any specifics as to the Composition.

I have known of a few tables not to care on UTH and LiR. Specifically, with LiR, one dealer said you couldn't show each other anything, but as far as they were concerned, you could talk about your hand. Of course, I don't think Comp-Dependent strategy would overcome the HE on LiR even with perfect knowledge, mainly because the knowledge doesn't actually change what the correct decision is very often.

I believe this topic has been discussed before. Knowing everyone else's cards doesn't help much and you're still playing at a disadvantage.

Quote: Ibeatyouraces

I believe this topic has been discussed before. Knowing everyone else's cards doesn't help much and you're still playing at a disadvantage.

I agree with that, but the knowledge can move the House Edge a little. I think UTH is a great example because, in order to make the final call bet, you want the dealer to have fewer than 21 cards that are able to beat you. In this instance, while it doesn't change the overall HE very much, the other players' hands could very easily make or break that decision.

I still disagree, it's VERY EASY for the dealer to qualify in TCP with three low cards. Ask any hole carder who's been completely frustrated when he plays hands like A,A,K after seeing a 3 only to have the dealer turn over a 2,4 with that 3!

Quote: Ibeatyouraces

I still disagree, it's VERY EASY for the dealer to qualify in TCP with three low cards. Ask any hole carder who's been completely frustrated when he plays hands like A,A,K after seeing a 3 only to have the dealer turn over a 2,4 with that 3!

That's where deck-composition comes into play. I think we're actually agreeing, here, while thinking that we disagree. My assertion is that a bunch of high cards in the hands of fellow players on the table very slightly increases the range of playable hands given the reduced likelihood of the dealer qualifying and/or beating you. We could both sit here and get into example hands all day, I imagine, but generally speaking, if a bunch of high cards are out, then the dealer is less likely to make a qualifying hand. When it comes to straights, I would say the probability of pulling a straight could increase, but not to the extent that winning with a high card decreases.

Remember, as the player, whether it's a high card (and better kicker, if a Queen) or it's a straight, you lose all the same, anyway. The only question that comes into play is the probability of the dealer beating you whilst qualifying, or not qualifying, and therefore, losing.

Anyway, I 100% agree that you don't have an advantage with even perfect collusion. I think that's been proven, somewhere. I could find it if I must. But, the potential for deck-comp-strat also decreased the HE a little bit.

Keep in mind, I'm talking stuff like Q-5-x as opposed to Q-6-4, minor decisions like that. I'm not saying that you start playing hands that wouldn't even qualify, were you the dealer. Although, there may be a few EXTREMELY rare cases in which you would even do that. Every single high card out, or something.

Quote: Mission146

Anyway, I 100% agree that you don't have an advantage with even perfect collusion. I think that's been proven, somewhere. I could find it if I must. But, the potential for deck-comp-strat also decreased the HE a little bit.

Either way, I know it's been discussed somewhere. Either here or in a book. I'll have access to Exhibit CAA later and I'll see if there is anything in there.

Quote: Ibeatyouraces

Either way, I know it's been discussed somewhere. Either here or in a book. I'll have access to Exhibit CAA later and I'll see if there is anything in there.

It's in CAA,I forget the numbers but ,basically it's worthless.

Quote: Hunterhill

It's in CAA,I forget the numbers but ,basically it's worthless.

Another problem is figuring out strategy adjustments based on what's seen. Making mistakes here probably will make it worse than better.