House edge of "Buying" the Ante Bet in 3 Card Poker

I'm trying to figure out the house edge in the following 3 Card Poker scenario:

In a 3CP game where the dealer shows one card, and Ante Pays w/ Any dealer non-Qualify. I am presented w/ the opportunity to "Buy" another players non-qualifying hand if they decide not to play. If dealer does not qualify at showdown, my Play bet pushes, I get the players original Ante Bet (that would have originally been surrendered), and the Ante payout from the house. Essentially, my 1 unit Play bet pays 2-1.

The probabilities of the dealer not qualifying is 30.41%. If I did my math correctly...
(2)(30.41%) + (-1)(69.59%) = -8.77% house edge.

QUESTION: How do I account for the dealer showing 1 card to my advantage? The above formula assumes all three dealer cards are unseen. Obviously, if the dealer card is Q or higher, I would not "Buy" for the dealer to not Qualify.

Any help would be appreciated.

Obviously you need to calculate the probability for not qualifying, giving the shown upcard.

So that's what I can't figure out, which calculation should be used...I'm terrible at math, I only pulled the above equation from a website.

I guess the equation I need is of the 69.59%, what is the probability that the upcard (1/3) will show a Q,K,A (12/52)?

Quote: BayAreaBear

In a 3CP game where the dealer shows one card...

Are you talking about hole-carding, or is there a variant of 3CP where the dealer exposes 1 card before the players make the "Play" decision? That seems like an incredible advantage to the player and I'd be interested to know all the rules and where this can be played.

Also, is this "buying another players hand" part of the game, or just an ad hoc offer you make? I ask because where I deal the players aren't supposed to share the contents of their hand with other players, so if that's going on that also seems like a big advantage for the player.

Quote: MonkeyMonkey

Are you talking about hole-carding, or is there a variant of 3CP where the dealer exposes 1 card before the players make the "Play" decision? That seems like an incredible advantage to the player and I'd be interested to know all the rules and where this can be played.

Also, is this "buying another players hand" part of the game, or just an ad hoc offer you make? I ask because where I deal the players aren't supposed to share the contents of their hand with other players, so if that's going on that also seems like a big advantage for the player.

I'm reasonably certain the OP's situation is thus.

Playing a 3CP game where the dealer is flashing, so you know the one of the dealer's hole cards, but this information is not obvious to the rest of the table. Given this situation, when would it be profitable to "buy" the hand of another player who would normally be folding, but you would want to play his hand based on your additional information.

Quote: BayAreaBear

I'm trying to figure out the house edge in the following 3 Card Poker scenario:

In a 3CP game where the dealer shows one card, and Ante Pays w/ Any dealer non-Qualify. I am presented w/ the opportunity to "Buy" another players non-qualifying hand if they decide not to play. If dealer does not qualify at showdown, my Play bet pushes, I get the players original Ante Bet (that would have originally been surrendered), and the Ante payout from the house. Essentially, my 1 unit Play bet pays 2-1.

The probabilities of the dealer not qualifying is 30.41%. If I did my math correctly...
(2)(30.41%) + (-1)(69.59%) = -8.77% house edge.

QUESTION: How do I account for the dealer showing 1 card to my advantage? The above formula assumes all three dealer cards are unseen. Obviously, if the dealer card is Q or higher, I would not "Buy" for the dealer to not Qualify.

Any help would be appreciated.

Hmmm.....

Let's first look at a worst-case scenario type situation in which you do not qualify and that player does not qualify. In such a scenario, both players have combined to remove six cards that would not qualify and the dealer (you know) has shown one card that does not qualify. This leaves 15 cards with which the dealer would automatically qualify and 45 total unknown cards in the deck if the dealer's shown card does not match any of the cards from either yourself or the other player. I say fifteen because the dealer could always pair up on his otherwise non-qualifying card.

If the dealer draws another card that fails to qualify him, then there will be (at most) 18 total cards which would then qualify the dealer, because he could now pair up the second card that was neither A-K-Q and didn't pair the first. Barring Straights/Flushes, the probability of the dealer not qualifying in a worst case scenario is:

(30/45) * (26/44) = 0.3939393939393939

Ok...now we have to account for the probability of a straight/flush and subtract it from the above result. Let's say that the dealer shows down an x of Clubs in this worst-case scenario and nobody has any other clubs. In that event, you would have 12 Clubs remaining in the deck of 45 cards and the dealer would need two of them, the probability of the dealer hitting a Flush would be:

(12/45) * (11/44) = 0.06666666666666666

Now you are at: 0.3939393939393939 - 0.06666666666666666 = 0.32727272727272727 probability of the dealer not qualifying in this worst-case scenario.

Finally, we must account for the probability of a straight. Here's another worst case scenario in which the dealer is open-ended, which is simply to say that he didn't show a Deuce-Three-Ten-Jack (Because a Queen beats you anyway). Let's say the dealer showed a Four of clubs and between you and the other player, there are no fours, Clubs, Deueces, Threes, Fives or Sixes. The dealer could make straights of 2-3-4, 3-4-5, 4-5-6:

The, "Order," of the cards is irrelevant, so the straight would require that the player definitely draw a 3 or 5, and then the player could draw the opposite of the 3/5, draw a Deuce to a Three or Draw a six to the five. The probability of a three/five is 8/45:

(8/45 * 8/44) = 0.03232323232323232

Now you are at: 0.32727272727272727 - 0.03232323232323232 = 0.29494949494949496 for the dealer not to qualify.

So, (2 * 0.29494949494949496) - (1 * (1-0.29494949494949496)) = -0.11515151515151511 or a HE of 11.5151% in this worst-case scenario.

Matching Card or 1 A-K-Q for player 1

However, if we knows that one of the player cards has the same value as the dealer's shown card, or that player one already has an A-K-Q then he has the A-K-Q, two cards to pair, and then (at most) three cards to pair his second card if the second card is not A-K-Q-4.

(31/45) * (27/44) = 0.42272727272727273 not to qualify that way for a difference of:

0.42272727272727273 - 0.3939393939393939 = 0.028787878787878806

Add that to the original, "No qualify," probability and:

0.028787878787878806 + 0.29494949494949496 = 0.32373737373737377

(2 * 0.32373737373737377) - (1 * (1-0.32373737373737377)) = -0.028787878787878695 or 2.87879% HE

Almost there!

Same Scenario, but Dealer has J

If the dealer card is a Jack, then the dealer effectively has only one way to make a straight because the Queen would beat you anyway.

(8/45) * (4/44) = 0.01616161616161616

Subtract that from the straight probability from before:

0.03232323232323232 - 0.01616161616161616 = 0.01616161616161616

0.32373737373737377 + 0.01616161616161616 = 0.3398989898989899

(2 * 0.3398989898989899) - (1 * (1-0.3398989898989899)) = 0.01969696969696977 or a 1.9697% Player Edge

Same Scenario, but Dealer has 10 instead of Jack

In this case, the dealer would specifically need J-9 or 8-9 to pick up the straight. The dealer must have the nine, of course.

(4/45) * (8/44) = 0.01616161616161616----Same thing.

Player 1 has two A-K-Q OR Player 1 has one A-K-Q + 1 Card of same Value as Dealer, OR Player 1 has one A-K-Q and Player 2 has one card of Same Value as Dealer

There would be thirteen ways for the Dealer to qualify with his first card, inclusive to Pairing or hitting A-K-Q:

(32/45) * (29/44) = 0.4686868686868687

The difference in DNQ percentage is 0.4686868686868687 - 0.3939393939393939 = 0.07474747474747478 based on that.

0.29494949494949496 + 0.07474747474747478 = 0.36969696969696974

That's a player advantage without even looking.

Conclusion Time

I could have screwed this thing up in any number of ways, and there may also be situations in which enough suits matching the dealer, or enough cards blockign a dealer straight would help prevent the dealer from qualifying, but generally speaking you would make this bet if:

1.) Your hand has at least 2 A-K-Q

2.) Your hand has at least 1 A-K-Q AND one card matching the dealer's card.

3.) Your hand has at least 1 A-K-Q AND Player 2 has at least one card matching the dealer's card.

4.) Your hand has only one A-K-Q, no cards match the dealer, BUT the dealer showed 10, Jack or Deuce.

5.) Your hand has only one card matching the dealer, BUT the dealer showed 10 or Jack or Deuce.

6.) Your hand does not have A-K-Q, or any card matching the dealer, but the other player has a card matching the dealer AND the dealer showed 10, Jack or Deuce.

Hopefully, someone will correct me if I have erred somewhere...

I'm not a math whiz, so I won't (can't?) comment on that, but in the beginning you refer to the players as having hands that don't qualify. FWIW, only the dealer can have a hand that doesn't qualify. The players can have hands they should fold if following the basic strategy for the game, but there is no qualifying criteria, other than the player places the "Play" bet, and the player may do so with any crap hand they like.

I'm also a little confused about the Conclusion section where in Option 1 you mention: Your hand has at least 2 A-K-Q. I'm puzzled by what you mean by this.

I still await the OP's clarification of whatever it is that's going on. I think AcesAndEights probably got it right, but I'd be interested to know if there is a variant being played somewhere that has the "rules" mentioned in the OP.

Quote:

I'm not a math whiz, so I won't (can't?) comment on that, but in the beginning you refer to the players as having hands that don't qualify. FWIW, only the dealer can have a hand that doesn't qualify. The players can have hands they should fold if following the basic strategy for the game, but there is no qualifying criteria, other than the player places the "Play" bet, and the player may do so with any crap hand they like.

I probably could have phrased that better, I simply meant any situation in which the hand for both players or Player 2 is worse than Queen-High. The reason that I used that for a worst-case scenario is because taking a potential Queen away from the dealer helps the player tremendously, so we needed to start from the premise that neither player has an A-K-Q or any card matching the dealer in value.

Quote:

I'm also a little confused about the Conclusion section where in Option 1 you mention: Your hand has at least 2 A-K-Q. I'm puzzled by what you mean by this.

He's talking about buying the hand of Player 2, so I am saying that if his (Player 1's hand) has a combination of two Aces, Kings, Queens that he should definitely do that because you've taken enough qualifying cards away from the dealer.

Quote:

I still await the OP's clarification of whatever it is that's going on. I think AcesAndEights probably got it right, but I'd be interested to know if there is a variant being played somewhere that has the "rules" mentioned in the OP.

It has to be hole-carding, because certainly if the dealer willingly gave the player knowledge of a card, there would be some off-setting rule mentioned by the OP.

I just cross-checked Ask the Wizard on WoO in the Q&A and he says to always buy it if the dealer flashes 2-J. If The Wizard says it, then that is absolutely right. I should assume that following my method would increase your advantage, though, but you would be buying less hands.

Quote: Mission146

He's talking about buying the hand of Player 2, so I am saying that if his (Player 1's hand) has a combination of two Aces, Kings, Queens...

This is where I get fuzzy. You seem to be saying the player has 6 cards. Perhaps I'm reading it wrong, it's been a long night.

Quote: Mission146

It has to be hole-carding, because certainly if the dealer willingly gave the player knowledge of a card, there would be some off-setting rule mentioned by the OP.

You'd think, but posters here make strange claims from time to time, so when trying to figure out the ambiguity I usually leave what's logical out because if they'd been logical in the first place there'd be no ambiguity.

Quote: MonkeyMonkey

This is where I get fuzzy. You seem to be saying the player has 6 cards. Perhaps I'm reading it wrong, it's been a long night.

No, I'm saying that he knows what six cards are. I'm assuming he is allowed to look at the other player's hand before he decides whether or not he wants to buy it. Is that incorrect? I've played 3CP maybe twice, lifetime, and was never presented with such a situation.

Quote:

You'd think, but posters here make strange claims from time to time, so when trying to figure out the ambiguity I usually leave what's logical out because if they'd been logical in the first place there'd be no ambiguity.

Let's not be mean! Judging from his verbiage, I think he simply didn't want to come out and say he's hole-carding.

Quote: Mission146

No, I'm saying that he knows what six cards are. I'm assuming he is allowed to look at the other player's hand before he decides whether or not he wants to buy it. Is that incorrect? I've played 3CP maybe twice, lifetime, and was never presented with such a situation.

It ranges from frowned upon to strictly not allowed. I suppose it's possible some places don't care at all, but given the advantage it would given the players it's hard to imagine that it's common anywhere.

Quote: Mission146

Let's not be mean! Judging from his verbiage, I think he simply didn't want to come out and say he's hole-carding.

Hard to understand why, there are some (many?) here that wear their hole-carding skills like a badge of honor.

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Quote: MonkeyMonkey

It ranges from frowned upon to strictly not allowed. I suppose it's possible some places don't care at all, but given the advantage it would given the players it's hard to imagine that it's common anywhere.

In that event, he would do it or not do it based exclusively on the cards in his hand compared to the dealer's upcard. That's with my method. The Wizard, again, says always do it if the dealer card is 2-J, so that is correct. I think it's just the right play in the overall sense, whereas, with my method, you know you have +EV on every individual play.

Quote: Ibeatyouraces

It's really not a skill but more of being aware. My suggestion is to NOT buy the hand whether its worth it or not.

I considered mentioning that and agree. If there was a telltale sign for the eye in the sky or pit crew that the dealer is flashing, buying a hand that someone else wants to throw away would be it!

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Yeah, if you had someone that was inconsiderate to other AP's, then maybe they could do it on a hit and run type of basis if they didn't care about the dealer inevitably being pulled or re-trained.

@Monkey and Aces,

Sorry for the late reply, in regards to the question of the dealer "flashing" or exposing an up card. There is a variation of 3CP in some local Bay Area Card Clubs that I play in that has the dealer openly show 1 up card prior to the player betting the Play. So for example if your hand is Q-3-2, and the dealers' up card is a Q, chances are you will not risk playing your hand.

Quote: MonkeyMonkey

Are you talking about hole-carding, or is there a variant of 3CP where the dealer exposes 1 card before the players make the "Play" decision? That seems like an incredible advantage to the player and I'd be interested to know all the rules and where this can be played.

Also, is this "buying another players hand" part of the game, or just an ad hoc offer you make? I ask because where I deal the players aren't supposed to share the contents of their hand with other players, so if that's going on that also seems like a big advantage for the player.

"Buying another players hand" is an ad hoc offer the casino allows you to do. I know in most casinos, it is usually one player per hand, in this particular casino, a player can bet on multiple hands as well.

I would not do it because of the increased attention. All the other players will inquire about your "strategy" and the pit boss may as well.

Quote: Mission146

Hmmm.....

Let's first look at a worst-case scenario type situation in which you do not qualify and that player does not qualify. In such a scenario, both players have combined to remove six cards that would not qualify and the dealer (you know) has shown one card that does not qualify. This leaves 15 cards with which the dealer would automatically qualify and 45 total unknown cards in the deck if the dealer's shown card does not match any of the cards from either yourself or the other player. I say fifteen because the dealer could always pair up on his otherwise non-qualifying card.

If the dealer draws another card that fails to qualify him, then there will be (at most) 18 total cards which would then qualify the dealer, because he could now pair up the second card that was neither A-K-Q and didn't pair the first. Barring Straights/Flushes, the probability of the dealer not qualifying in a worst case scenario is:

(30/45) * (26/44) = 0.3939393939393939

Ok...now we have to account for the probability of a straight/flush and subtract it from the above result. Let's say that the dealer shows down an x of Clubs in this worst-case scenario and nobody has any other clubs. In that event, you would have 12 Clubs remaining in the deck of 45 cards and the dealer would need two of them, the probability of the dealer hitting a Flush would be:

(12/45) * (11/44) = 0.06666666666666666

Now you are at: 0.3939393939393939 - 0.06666666666666666 = 0.32727272727272727 probability of the dealer not qualifying in this worst-case scenario.

Finally, we must account for the probability of a straight. Here's another worst case scenario in which the dealer is open-ended, which is simply to say that he didn't show a Deuce-Three-Ten-Jack (Because a Queen beats you anyway). Let's say the dealer showed a Four of clubs and between you and the other player, there are no fours, Clubs, Deueces, Threes, Fives or Sixes. The dealer could make straights of 2-3-4, 3-4-5, 4-5-6:

The, "Order," of the cards is irrelevant, so the straight would require that the player definitely draw a 3 or 5, and then the player could draw the opposite of the 3/5, draw a Deuce to a Three or Draw a six to the five. The probability of a three/five is 8/45:

(8/45 * 8/44) = 0.03232323232323232

Now you are at: 0.32727272727272727 - 0.03232323232323232 = 0.29494949494949496 for the dealer not to qualify.

So, (2 * 0.29494949494949496) - (1 * (1-0.29494949494949496)) = -0.11515151515151511 or a HE of 11.5151% in this worst-case scenario.

Matching Card or 1 A-K-Q for player 1

However, if we knows that one of the player cards has the same value as the dealer's shown card, or that player one already has an A-K-Q then he has the A-K-Q, two cards to pair, and then (at most) three cards to pair his second card if the second card is not A-K-Q-4.

(31/45) * (27/44) = 0.42272727272727273 not to qualify that way for a difference of:

0.42272727272727273 - 0.3939393939393939 = 0.028787878787878806

Add that to the original, "No qualify," probability and:

0.028787878787878806 + 0.29494949494949496 = 0.32373737373737377

(2 * 0.32373737373737377) - (1 * (1-0.32373737373737377)) = -0.028787878787878695 or 2.87879% HE

Almost there!

Same Scenario, but Dealer has J

If the dealer card is a Jack, then the dealer effectively has only one way to make a straight because the Queen would beat you anyway.

(8/45) * (4/44) = 0.01616161616161616

Subtract that from the straight probability from before:

0.03232323232323232 - 0.01616161616161616 = 0.01616161616161616

0.32373737373737377 + 0.01616161616161616 = 0.3398989898989899

(2 * 0.3398989898989899) - (1 * (1-0.3398989898989899)) = 0.01969696969696977 or a 1.9697% Player Edge

Same Scenario, but Dealer has 10 instead of Jack

In this case, the dealer would specifically need J-9 or 8-9 to pick up the straight. The dealer must have the nine, of course.

(4/45) * (8/44) = 0.01616161616161616----Same thing.

Player 1 has two A-K-Q OR Player 1 has one A-K-Q + 1 Card of same Value as Dealer, OR Player 1 has one A-K-Q and Player 2 has one card of Same Value as Dealer

There would be thirteen ways for the Dealer to qualify with his first card, inclusive to Pairing or hitting A-K-Q:

(32/45) * (29/44) = 0.4686868686868687

The difference in DNQ percentage is 0.4686868686868687 - 0.3939393939393939 = 0.07474747474747478 based on that.

0.29494949494949496 + 0.07474747474747478 = 0.36969696969696974

That's a player advantage without even looking.

Conclusion Time

I could have screwed this thing up in any number of ways, and there may also be situations in which enough suits matching the dealer, or enough cards blockign a dealer straight would help prevent the dealer from qualifying, but generally speaking you would make this bet if:

1.) Your hand has at least 2 A-K-Q

2.) Your hand has at least 1 A-K-Q AND one card matching the dealer's card.

3.) Your hand has at least 1 A-K-Q AND Player 2 has at least one card matching the dealer's card.

4.) Your hand has only one A-K-Q, no cards match the dealer, BUT the dealer showed 10, Jack or Deuce.

5.) Your hand has only one card matching the dealer, BUT the dealer showed 10 or Jack or Deuce.

6.) Your hand does not have A-K-Q, or any card matching the dealer, but the other player has a card matching the dealer AND the dealer showed 10, Jack or Deuce.

Hopefully, someone will correct me if I have erred somewhere...

Mission, thank you for your detailed breakdown, I haven't had a chance to go through it the whole way, as I am admittedly terrible at math, but I certainly will as I appreciate your efforts to help me break this situation down to a mathematical level.

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Thank you all for contributing to this thread, and I again apologize for not responding earlier. To clarify my scenario, where I play 3CP at Livermore Casino in the Bay Area, the dealer openly shows 1 up card prior to the payer deciding to make the play bet. So, if player has a K high, and the dealer up card is an Ace, the player will automatically fold.

Secondly, a player can bet on multiple hands, in this casino, 7 player hands are dealt (on a full table). So for instance if I am playing on seat 1, and seat 7 is on a hot streak, I can bet on seat 7's spot as well.

In regards to the my original post, I have confirmed w/ the pit bosses at Livermore that from the rail (meaning I am not even a seated player), I can make a play bet on any seated players hand who is folding, thus "buying a players hand".

For example, if the dealer up card is a J, and a player hand is a non-qualifyer. The player has decided to fold and surrender his 1 unit Ante bet. I can from the rail, put up the matching 1 unit Play bet for that hand. IF, the dealer ends up w/ a non-qualifyer as well, my Play bet is a Push, I get the original Ante bet (of the Player) and the 1-1 payout on the Ante bet (from the Casino) for the dealer not Qualifying. Essentially, I am getting 2-1 on my Play bet if dealer does not qualifying, and only losing 1 unit if Dealer does.

Quote: Ibeatyouraces

See the Wizards section on California Three Card Poker over on WizardofOdds. If the version you describe is the same where your non qualifier still has to beat a dealer nonqualifier, I wouldn't buy the folded hand.

In the casino where I play, any dealer non-qualifier is a loser regardless of player hand.

Player 2-3-5 > Dealer J-10-8

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Quote: Ibeatyouraces

Essentially this seems the same as playing with a flashing dealer. I'm guessing there is no ante bonus on a straight or better.

Correct, no Ante bonus for straight or better.

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