Card Combinations

It is important to evaluate the number of card combinations when trying to accurately estimate an opponent’s hand. For example, you can be dealt pocket aces six different unique ways. You can get dealt the ace of spades with each of the other three aces: AsAc AsAd AsAh

Then you can get dealt the ace of hearts with each of the other three aces: AhAd AhAc AhAs

But wait. See AhAs above? We already counted it in the ace of spades combinations. So we exclude AhAs because the ORDER of the cards do not matter in hold’em. Of course when you get dealt any two cards, you can receive the first one first or the second one first, it doe not matter. What this means is there’s twice the card combos IF orders matters, but it does not for poker.

So we take out the duplicate AhAs: AhAd AhAc

Then the ace of clubs, not including the two AsAc and AhAc combos we already have above: AcAd

And notice the ace of diamonds has already been paired up with each other ace above. We have a grand total of 6 unique ways you can be dealt pocket aces; six “combinations”.

Now we can go through the same formula to determine that for unpaired hands, there are sixteen combos of each. For example, there are 16 unique ways we can be dealt AK. Note that 12 of those combos are unsuited AK, and 4 are suited, which is important.

For an example how we can use and abuse our knowledge of hand combinations to profit in poker, imagine a simple scenario where our opponent bets. He either has AA or AK. We can beat AK, but lose to AA. Now in combos we know that there are 6 combos of AA and 16 of AK. This means he wins 6 times for every 16 times he loses. Since we are getting pot odds of 1:1, card combos indicate a call for us to be very profitable, since we win 16 times while losing only 6 times. That is, when we call his bet we will win 72.7% of the time. With pot odds of 1:1, we only need to win 50% of the time to break-even.

When can we apply this knowledge in hold’em to make money?

Example

Here’s a common situation where our knowledge of card combinations allow us to make a profitable play. We are playing $0.50/$1 NL, with $100 stacks.

It folds to the button and he raises to $4. The SB folds, and we are in the BB with QQ. How lucky! We make a standard re-raise to $16 (at least you SHOULD be re-raising here!). He then shove all-in for $100.

It’s $84 for us to call. The pot is now our $16 reraise plus his $100 shove; $116 (give or take a bit for rake and the SB).

As usual, first thing’s first: calculate our pot odds. We need to fold, which is 0 Expected Value (EV), if we fold we lose nothing more, plus profit nothing. Now, if calling is positive EV then we should call. If calling is negative EV we should fold. There are external reasons where you might choose one play or the other when the EV is close to 0 but let’s keep this simple.

Pot odds: 116:84 or 1.38 to 1. Converting to a percentage, we take 1.38 to 1 and change it to 1/2.38. That is, we need to win 1 time out of every 2.38 times to break-even (0 EV).

1/2.38 = 42%. If we call, we need to win 42% of the time to break-even.

And this is where card combinations come into play.

Say this guy is a maniac and he does this with every hand he is dealt. He could do this with AA, or with 72o. He didn’t even look at his hand. His hand is random. How does QQ hold up against a random hand?

Putting the hands into Pokerstove:

• equity win tie pots won pots tied
• Hand 0: 79.925% 79.63% 00.29% { QQ }
• Hand 1: 20.075% 19.78% 00.29% { random }

This means we win 79.9% of the time when we call against his random hand. Since 79.9% is greater than 42%, we have a very clear, profitable call.

Unfortunately, players that shove all-in with a random hand aren’t very common. What’s a more realistic range? Now let’s assume our opponent is good predictable player. In fact he is so predictable you know he will raise, and re-raise all-in pre-flop like this with only 3 hands: AA, KK and AK both suited and unsuited. His range is: { KK+, AKs, AKo }

We have about a 18.5% chance of beating AA, and 18% chance of beating KK, a 57% chance of beating AKo and a 54% chance of beating AKs. Since we have no information to determine which of these hands he has (since he plays them all the same way in this scenario) we need to combine them all and do the math against his range.

He has 6 combos of AA, 6 of KK, and 16 of AK.

We can use Pokerstove to enter this range, and have it do the math:

• equity win tie pots won pots tied
• Hand 0: 39.857% 39.64% 00.22% { QQ }
• Hand 1: 60.143% 59.92% 00.22% { KK+, AKs, AKo }

Our equity dropped to 39.9%, just a little less than the 42% we require to break-even. If we are so sure this player will only shove in this spot with AA, KK and AK, we should fold, as calling will lose us money in the long-run.

This is black and white; a play is either profitable or not. The pot odds are defined, and our equity is calculable. The only thing that isn’t defined is the range of hands our opponent has. That’s where experience comes in. The better you can accurately estimate your opponent’s hand range when he makes a certain play, the more profit you will make.

Okay, that’s a basic overview of card combinations and how you can use them to make money, let’s get into a complex situation that happens fairly regularly against good players, where we use our knowledge of hand ranges and card combos to attack a pre-flop raiser.

Attacking a UTG Pre-Flop Raiser in 6-max No Limit

The majority of the time a good standard 22/18 solid Tight-Aggressive (TAG) player at a 6-max NL cash game will raise if he is the first one in the pot. While we defend against this pre-flop aggression with some re-raises with strong hands, we sometimes cold-call the raise and see a flop; typically with small to medium pairs and suited connectors.

While we often miss the flop and face a seemingly inevitable continuation bet from the pre-flop raiser – there are times when we can bluff-raise profitably. Here are a couple profitable spots where you add this play to your game, making it more aggressive and unpredictable.

This scenario is designed off a very common situation. A good TAG raise UTG, and we call. The flop comes with a high card, a couple low cards and fairly dry. He continuation bets the flop, and we bluff raise.

The Scenario

A standard TAG player at a 6-max NL cash game raising UTG will have a range something like:

{ 22+, ATs+, KQs, QJs, JTs, ATo+, KQo }

The hand range for this UTG pre-flop raise is: all pairs, big aces, KQ and some big suited connectors. This range works out to 12.5% of his initial hands. The rest of the time he folds.

Since pairs are extremely profitable in NL, it’s become industry standard amongst winning players to open-raise all pairs in all positions at 6-max tables. However, when in early position, especially UTG, where a tight opening range is necessary, the pairs out-weigh the rest of the range and it opens an opportunity for the astute player to attack the UTG pre-flop raiser on certain boards.

Many players understand now it’s often profitable to bluff-raise the continuation bet on a dry board with a single unconnected high card, because it’s very hard for the initial preflop raiser to continue facing a raise. This example expands on this idea, using similar principles to attack an opponent when his range is tightest.

The UTG TAG player raises and we cold-call. If the board comes Axx or Kxx, there is often a situation where it is profitable to bluff-raise the TAG’s continuation bet.

For example, if the board comes Axx rainbow (no flush draw), the hot cold equity for this range versus Ax:

• Board: Ac 4d 2s
• equity win tie pots won pots tied
• Hand 0: 29.454% 28.05% 01.40% 299963 14957.00 { 22+, ATs+, KQs, QJs, JTs, ATo+, KQo }
• Hand 1: 70.546% 69.15% 01.40% 739323 14957.00 { A3o }

Which means that if it went to showdown, the initial 12.5% preflop raising range of { 22+, ATs+, KQs, QJs, JTs, ATo+, KQo } would win only 29% of pots against a weak ace.

On a flop of Ace-rag-rag (like A94 or A62)… the majority of the time (probably at least 2/3 of the time) the initial preflop raiser cant beat the weakest ace.

This is because of the card combinations in UTG’s open-raising range, typical ranges (as shown above) just doesn’t have that many Ace-x hands. Let’s take a more detailed look. On an Axx flop, three of the 13 pocket pairs flop sets. The other ten are behind a pair of aces.

• Sets: 9 combos total
• Other pairs: 60 combos
• Ax hands: 48 combos
• KQ: 16
• QJs, JTs: 8

57 hand combos flop a pair of aces or better, but 84 do not.

Now if he bets the pot, and we raise a standard 3x bet size, we are risking 3x to win 2x. To break-even on a straight bluff we need to take down the pot >60% of the time. So if he continuation bets 100% of the time he pre-flop raises UTG with a range near that, a raise on a Axx flop is profitable, with any two cards! Note that we are ignoring the fact that we will often have some equity, and it’s not often we will be on a pure bluff. Even raising with a small pocket pair, we likely have 2 outs to turn a set.

The numbers are similar for Kxx flops.

When there is a flush draw, the villain gains ~2% points. Additionally, we need to be more careful when raising boards with a flush draw because our opponent is more likely to call or reraise our raise because he will put a flush draw in our range. He will be less likely to fold a pair.

On Qxx flops, the villain’s hot-cold jumps to ~35%. This is because Q is more common in his range, and he is more likely to have an overpair (AA+KK vs. just AA)

So bluff-raising on Axx and Kxx flops against a UTG pre-flop raiser who continuation bets the flop a high majority of the time (say 90%+) is a break-even or profitable play even if our cards were blank.

Now, most players aren’t as robotic. Many good players will not continuation bet the flop after being the pre-flop raiser. Of course he range might not be as wide or as narrow. So if these factors mean it might make a play like this close to break-even or marginally profitable, then why do it at all?

Note I ignored the fact that we will always have some equity in the hand, that is our cards aren’t blank! We might have as little as 2 outs drawing to a set, or a gut-shot or something, but chances are we have at least 10-20% equity on the flop. That, when considering we often will have position, will add significantly to our profit.

Well, sometimes when they check to you, they will just give up. That’s good for you. Your aggressive defense will force the original pre-flop raiser to have to adjust by just letting you have some pots.

Once you add in that you sometimes will raise with a legit hand, and that you likely have some equity regardless, it is a profitable and extremely tough play for the opponent to exploit. You don’t want to fall into the routine trap when facing a TAG raiser’s continuation bet of raising your monsters, calling you marginal pairs and draws, and folding when you miss. While of course, you need to base your game fundamentally on folding your bad hands and putting in more money the stronger you hand gets; you need to add in a dose of mixing it up against good thinking players. Occasionally, you can win monster pots by throwing the opponent a curveball, but also pick up some extra pots in spots that are very tough for him to counter-act.

Your success with this play will depend on how accurate you can estimate his early-position pre-flop raising range, combined with attacking the various flops that maximize you bluff’s chance of success. Also, don’t forget the recent history. If you’ve been playing tight, your raise will certainly get more respect and thus have a greater chance of success, however if you’ve been splashing around lately and more importantly maybe have been pushing the villain off some hands recently – you must take that into consideration.

Assumptions:

6-max NL cash game. That a TAG player raises 12.5% of hands UTG which is a range of: { 22+, ATs+, KQs, QJs, JTs, ATo+, KQo }, that he continuation bets a Axx or Kxx flop close to 100% of the time. That he will fold hands worse than Top Pair to a flop raise. We are on a pure bluff with 0% equity.

To talk about the thread and read commentary, go here: http://www.604poker.com/phpBB2/viewtopic.php?f=6&t=12750

- Pete