# Discount Gambling

## Ultimate Three Card Poker (Face Up)

Posted in three card poker by stephenhow on July 23, 2012

Thanks to reader Jet for pointing out this new ShuffleMaster ™ game. I ran a quick analysis this morning to check his numbers, and of course to check for any collusion possibilities. An optimal strategy yields a -3.8807% return, and we pretty much came up with the same basic strategy (differences arise from suit assumptions). You should 3x raise all pairs or better, and 1x call according to the following table.

An ideal collusion simulation with 6 confederates yielded no practical gain π¦ This probably makes sense, because you can’t really take advantage of collusion information when they limit 3x raises to pairs. Perhaps you can get a slight edge on some 1x/fold borderline cases, but it doesn’t amount to anything significant.

Jet has a hole-carding strategy that yields 9%, which you can contact him for.

Ultimate Three Card Poker (Face Up)
Dealer Upcard Min Calling Hand
2 98x
3 Txx
4 T4x
5 T5x
6 T6x
7 T7x
8 T6x
9 Txx
T T84
J J84
Q Q84
K K84
A A85

## Three Card Hold’Em @ Golden Nugget, Las Vegas

Posted in three card hold'em, three card poker by stephenhow on March 25, 2012

Earlier this month, I saw a new 3-Card Hold’Em poker carnival game at the Golden Nugget in Las Vegas. I wrote down the rules, and finally got around to analyzing the game today. Coincidentally, I found a reader of this blog playing the game at the Nugget. We were joking about the game, when he flops the nut straight (he has AK, and the Flop is a Q). So he bets the Turn and River blind, and the board comes runner-runner spades. I figure he’s ok, because a straight beats a flush in 3-card poker. Then the dealer turns up her hole cards, and uses both of them to make a straight flush. Unbelievable.

### Rules

The game is played between the dealer and each player. Each player and the dealer receives two hole cards, and combines them with the community cards on the “board” to make their best 3-card poker hand. You do not need to use any of your hole cards to make a hand (i.e., you may “play the board”).

1. Player posts an Ante before the start of the hand.
2. The Player and the Dealer are each dealt two hole cards.
3. The dealer turns up the Flop card on the community board.
4. The Player decides whether to 1x bet the Flop, or to fold his hand and lose his Ante.
5. The dealer turns up the Turn card on the community board.
6. The player decides to either 1x bet the Turn, or check.
7. The dealer turns up the River card, then turns up his hole cards.
8. The Dealer requires a pair of 4’s or better to qualify, else the Antes push.
9. All remaining bets receive even-money action against the Dealer hand.

### Strategy

I constructed a reasonably simple strategy, and was very surprised to see it simulate at only a -0.60% EV! This is pretty good for a carnival game, which usually has between a 2.5% to 3.5% house edge. My simple strategy isn’t very optimal, and I estimate the actual house edge is very close to 0. (The error EV of my simple strategy simulates around 0.5%, so an optimal strategy would yield close to 0 EV.)

On the Flop, you should always make the 1x bet (never fold your Ante).

On the Turn, use the following table to decide when to bet your hand, else check. You should only bet a pair under the stated conditions, and when your hand beats the board by more than just kickers, except when betting the nut flush draw.
(Glossary: “scare straight” means the board has connected cards; “scare flush” means the board is suited; “good straight draws” (GSDs) mean the number of River card ranks that make you a straight, excluding the “idiot end” straight draws. For example, if your hand is Kc 6h and the board is Qd 7s, then your 3 GSDs are {A,J,5}. Note that an 8 is not a GSD, but an “idiot-end” draw, because a dealer 9 would beat your 8-high straight. Basically, a GSD is an out that makes a straight which can’t be beat by a single dealer hole card.)

Hand on Turn Turn Bet Requirements
Straight or better Always bet.
Flush If scare straight AND scare flush board, only bet 4th nut flush.
One Pair If not scare straight nor scare flush board, bet your pair of Jacks or better, else
if not scare flush, bet your pair with 2 or more GSDs, else
if not scare flush, nor scare straight, bet your pair with 4th nut flush draw or better, else
if not scare flush, bet your pair of 6’s or better, if you have any GSDs and any flush draws, else
if the board is paired, bet your nut flush draw.
No Pair If not scare straight and not scare flush, bet 4 GSDs and 9th nut flush draw or better, else
if not scare straight and not scare flush, bet 2 GSDs and two flush draws with a 7th nut flush draw or better.
Simple Strategy for Turn Bet.

This simple strategy decisions are occasionally wrong by more than 10% of the Turn bet, but overall, it works out well enough. The mistakes are pretty obvious when you look at them, it’s just not worth optimizing the strategy unless you want to play it seriously. The game will probably be gone in a few weeks anyway, I just wanted to see what a reasonably strategy would yield.

### Bonus Bets

The Straight-or-Up bonus bet has a 4.49% house edge, which isn’t too bad, as far as side bets go.

Hand Payout Frequency Probability Return
AKQ-suited + pair 100 276 0.000106 0.010620
AKQ-suited 40 4428 0.001704 0.068150
Straight Flush 10 49628 0.019015 0.190953
Trips 9 58848 0.022643 0.203787
Straight 1 569268 0.219037 0.219037
Others -1 1916512 0.737415 -0.737415
Total 2598960 1.000000 -0.044868
Straight-or-Up Final Hand Side Bet

The Pair-or-Suited side bet on your two hole cards is a little better, costing only about a 4.8% house edge.

Hand Payout Frequency Probability Return
AK-suited 30 4 0.003017 0.090498
AA 20 6 0.004525 0.090498
KK 10 6 0.045249 0.045249
Pair 4 66 0.049774 0.199095
Suited 1 303 0.232278 0.232278
other -1 936 0.705882 -0.705882
Total 1326 1.000000 -0.048265
Pair-or-Suited Hole Card Side Bet

## Mini-Tex 3-Card Hold’Em @ Santa Ysabel Casino, CA

Posted in mini-tex, three card poker by stephenhow on August 3, 2011

It was bound to happen sooner or later that someone would invent a 3-Card Hold’Em carnival game. I first heard about this game from the Wizard Of Odds last week. Since they actually have this game at my nearby Santa Ysabel Casino (min \$2 Ante, \$1 Bonus), I thought I’d figure out a strategy, and give it a try.

The game is pretty simple, and the Wizard lists the full rules. Like Hold’Em, each player has two hole cards, and combines them with a community board to make his best 3-Card Poker hand. The board consists of the Flop (2 cards), and the River (1 card). To form a hand, the player must use at least one of his hole cards (i.e., he cannot play the board).

### Rules

1. The player bets an Ante to start the hand.
2. Each player and the dealer receive two face-down hole cards.
3. The player decides to either wager the 1x Flop bet, or fold his hand and Ante.
4. Two community cards are dealt to the board (the “Flop”).
5. The player decides to either wager the 1x River bet, or fold his hand and all his bets.
6. One last community card is dealt to the board (the “River”).
7. The player decides to either wager the 2x Play bet, or fold his hand and all his bets.
8. At showdown, the dealer must have a pair to qualify, else the Ante, Flop, and River bets automatically win, and the Play bet pushes.
9. If the dealer does qualify, then all bets receive even-money action against the dealer hand.

### Simple Strategy

The theoretical house edge for this game is 3.22%. The simple strategy described here returns 4.0%, which isn’t bad, considering the complexity of draws, and possible dealer hands against various boards. The simple strategy is extremely easy to remember. The game is fun and easy to play, because you only fold really bad hands (they’re rare).

#### Flop Bet

According to the Wizard of Odds, the player should play everything except 22 and 25o. Using my simplified strategy, you should also fold 26o.

#### River Bet

To see the river, you should 1x bet any of the following hands (fold everything else):

• Any pair (that beats the board), or better.
• Bet two or more draws (e.g., two flush draws, or two straight draws, or a straight draw and a flush draw).
• Against a paired flop, bet any straight draw, any flush draw, or 3rd nut kicker.
• Bet a lone gutshot against an offsuit, gapped flop.
• Bet a single flush draw against a gapped flop.

* “gapped flop” means the flop isn’t a pair, or connected (e.g., 6-7 is connected, while 6-8, 7-2, A-6 are gapped)

#### Play Bet

To go to showdown, you should Play (2x) any of the following hands (fold everything else):

• Mid pair or better.
• If board is not paired and all different suits, play anything (i.e., bet that dealer doesn’t qualify).
• If the board shows a flush (3 of same suit), and there are no one-card straights, play anything.
• If board is paired, bet your kicker if there are less than 21 cards that beat you (the lowest possible kicker is a 9 or T; it increases by about one level for each one-card straight).
• Bet bottom pair unless both a one-card dealer flush and a one-card dealer straight are possible.

### Examples

You have 2s3c and the flop is 5dKh. You have a lone gutshot draw (you need a 4 on the river). You bet the gutshot draw, because the board isn’t suited nor connected.

You have 8d5h and the flop is Kh2c. You only have a flush draw. You bet the flush draw, because the board is gapped.

You have garbage. The board is 5d6h7c. You bet 2x hoping the dealer doesn’t qualify, because the board is rainbow and not paired. (There are 5 possible one-card straights, but that’s ok.)

You have 9h2c. The board is 6s6dAh. You can bet 2x your 9, since the board is rainbow, and there are no one-card possible straights.

You have Kh2c. The board is 6s6dAd. You can bet 2x with the K, since there are only 19 cards that beat you.

You have Qh2c. The board is 6s6d5c. You can bet 2x with the Q, since there are 21 cards that beat you.

### Collusion

I can’t believe this game doesn’t lend itself to collusion between players sharing hole card info. I thought there’d be a lot of opportunities for collusion, like the Flop bet decision, if confederates hold your adjacent cards (very important for straights). Or, at showdown, when you call with nothing against an unpaired, rainbow board. But my analysis shows confederate hole card info doesn’t help much at all π¦ I really thought that card info would be important, because in 3-card poker, one card often makes a hand.

### I Played It!

I went out to Santa Ysabel Casino, and played the game for a few hours. It was really fun! The minimum Ante bet is only \$2! So, the most you could lose in a hand is \$10, and the house edge is only about \$0.08/hand. The strategy is really easy to implement. You only fold if you have absolutely nothing, so it’s pretty easy to play (sometimes you even 2x Play bet nothing!). I usually only looked at one card to make my Flop and River bet decisions. Sometimes, I didn’t even need to see both cards to make a 2x Play bet decision. It was a lot of fun. The only time I ever needed to think was on the Play bet decision with kickers against a paired board (subtracting from 6th nut kicker for one-card flushes and straights).

While it’s easy to play the strategy, most of the other players probably have difficulty figuring out the right move on the fly. So, you’ll enjoy being the expert at the table.

I recommend the game, because it’s fun, and you’re usually betting 5 units to either win even money, or 3-to-5 if the dealer doesn’t qualify. That makes the house advantage per average bet fairly low (4.0%/4.6 units) < 1%.

## Face Up β’ Three Card Poker @ Casino Pauma, CA

Posted in three card poker by stephenhow on July 30, 2011

They’ve added a new Three Card Poker table to Casino Pauma, and I got excited when they said they exposed a dealer card. However, unlike the Three Card Poker at Ocean’s 11, they don’t pay the Ante if the dealer doesn’t qualify π¦ They remove the Ante bonuses, so while you get to see a dealer card, it still ends up being worse than regular Three Card Poker. The Wizard Of Odds posts an optimal strategy that yields a 4.3% house edge, while the simple strategy below returns a slightly worse 4.6% house edge. All-in-all, I’d rather have the regular game and it’s more reasonable 3.4% house edge.

Face Up ™ Three Card Poker Basic Strategy
Hand Decision
Losing hand Fold
Pair or Better Play
Ten-high or less Fold
Jack-high Play if your 2nd card higher than the dealer upcard; else Fold.
Queen-high Play against a lower dealer upcard, and play Q-9 against a Queen upcard; else Fold.
King-high Play against a lower dealer upcard, and play K-9 against a King upcard; else Fold.
Ace-high Play against a lower dealer upcard, and play A-9 against an Ace upcard; else Fold.

I looked into the benefit of collusion on Jack-high decisions, and found only a few tenths of a percent improvement. Unlike Caribbean Stud, where collusion improves the return by 6.5%, the effect of dealer qualifying is not that dramatic in Face Up Three Card Poker. So, there’s no practical point of colluding here.