Counting Cards

When people who do not understand poker ask me about it, they usually say things such as “You must be good at counting cards”, they associate gambling at cards with card counting. This is usually because they have heard of card counting at Blackjack. Basically to card count at Blackjack you assign a point value to cards depending on their ranks and this tells you how many high cards are left in the deck. The more high cards there are then the more chance that the dealer will bust and the odds can actually be in your favour if the deck is heavy with high cards. This means you can make a short term +EV play by betting more when the time is right, because of this casinos frown on card counting.

The film "21" tells the story of a group of MIT students attempting to count cards at Blackjack.

I don’t want to talk about card counting in Blackjack (Watch the film 21 if you want to see card counting in action), but when I am asked about card counting, it is easy to be dismissive and say there isn’t a need to count cards in hold-em, after all it is a game of non-perfect information (we don’t know what our opponents cards are) there are only 5 known community cards and the 2 we have in our hand that we know for certain. However despite not having perfect information, we can still utilise what information we do have to card count in selective ways when playing Hold’em. Whenever we have to make a decision in poker, we should try to put our opponent on a range of hands, the more accurately we can do that then the better our decision making will be and in the long run the more money we can expect to make.

Blockers

The easiest and most useful way of “counting” is to look at your cards and those on the flop and see how this might affect your opponents range and their chances of hitting their hand. Let’s say we hold AhJs and the flop is Ac 9h 3h and we are heads up against one opponent. What is the chance they have the nut flush draw? This isn’t a difficult question to answer and it doesn’t involve calculating any ranges, the answer should be really simple. If you have not worked the answer out yet go back and read the question, paying particular attention to the cards and their suits. Don’t read on until you have the answer, once you have the answer you will know it’s the right one.

Clue: Don't think about your opponent's hand at all, look at the cards we have and work out how this changes the odds for our opponent holding the nut flush draw.....

The answer is 0%! Since we have the Ah they cannot have the nut flush draw…in order for them to have the nut flush draw they must have Axhh, but they cannot have this as we already have the Ah! This can be really useful in some situations, say for example we raised pre and they called from the SB, and they check raise our continuation bet. If we have notes of them that say they would only check-raise with 2 pairs or better or the nut flush draw (i.e. they would just check call with one pair hands or flush draws that aren’t the nut flush draw) then in this scenario when we get check raised we can easily work out that we are beat and assuming we don’t have the correct odds we can fold. If instead of the Ah we have the Ad then this becomes a trickier spot as they villain might have Ah6h for example as that would be in their check raising range according to our notes.

It is also less likely they have two pair here as well when we hold any ace as it is less likely they have A9 or A3 and they probably don’t have 93. All of a sudden their range looks a lot like a set when we hold the Ah, whereas it is a bit wider if we hold another ace and wider still if we don’t have an ace at all.

The above example is a bit contrived as it assumes the villain plays a certain way and we know this, however even if we don’t know that for certain, clearly their semi bluffing range is smaller when we have the Ah than when we don’t, most opponents are more likely to play Axhh this way than a lot of the other xyhh combos (how many would call pre with J2hh for example then check raise the flop? It’s certainly less than would play A2hh that way). This means their range is usually skewed more towards value than semi bluffs.

Another scenario might be where we have AdKd and the flop is Ah9h3s. Although the Ah is om the board, it still means your opponent doesn’t have it. Although they may have the nut flush draw with Kxhh, the range of Kxhh they would play preflop is likely less than Axhh (e.g. more players would call pre with A2hh than K2hh) so we can still skew their range in a similar fashion to that above, although we should note hands like KQhh and KJhh are more likely in their c/r range than in the first example.

The lesson here is to pay attention to any cards you have that may affect the range your opponent has, which leads me nicely onto the next point.

Combinatorics (EV)

Now let’s suppose we have AKs and a player has opened, we have 3 bet them and now they 4 bet shove for around 35bb (which is the same stack we have). Our notes tell us they always have AK or QQ+ here, let’s assume the villain always 4 bet shoves these hands and nothing else. Given this we can work out our EV in the hand, to do this we can work out our EV against each hand and take the average.

                Our EV against AK = 50%

                Our EV against QQ = 46%

                Our EV against KK = 34%

                Our even against AA = 12%

So our average EV = (50 + 46 + 34 + 12)/4 = 35.5%.

However, there is a flaw in this calculation; can you see what it is?

The flaw is that when we took the average EV we assumed each of those 4 hands is equally likely, when in fact they aren’t! We forgot to take account of the effect of our blockers on the combinations of these hands available, let me explain further.

There are 4 aces in the deck, spades, diamonds, hearts and clubs. We can get dealt any of those 4 aces and in any order. Note that getting dealt Ah Ad is the same as Ad Ah. So how many combinations of these 4 aces are there? Well let’s work it out:

1. As Ad
2. As Ac
3. As Ah
4. Ad Ac
5. Ad Ah
6. Ac Ah

There are 6 combinations of Aces when none of the aces are known to be elsewhere. 

For the record there are 1,326 combinations of starting hands in holdem, each of which are equally likely. The chances of being dealt aces preflop is the number of combinations of AA (6) divided by the total possible combinations of any two cards (1326) so 6/1326 = 0.45%

Now let’s go back to our example where we have the As, how many combinations of Aces are there? Well Combo 1, 2 and 3 can no longer be valid as they have the As in so there are now only 3 combinations left. This holds true for any pair so there are also only 3 combinations of KK since we hold the Ks. However we know nothing about any Queens, so there are still 6 combinations of QQ. This means our opponent is just as likely to have QQ as they are to have KK or AA combined!

Now let’s consider the combos of AK (given that we have AKss):

1. Ah Kh
2. Ah Kd
3. Ah Kc
4. Ad Kh
5. Ad Kd
6. Ad Kc
7. Ac Kh
8. Ac Kd
9. Ac Kc

There are an additional 7 combos if we ignore the fact we have AKss:

1. As Kh
2. As Kd
3. As Kc
4. As Ks
5. Ad Ks
6. Ah Ks
7. Ac Ks

Ok, so how does that affect our EV calculation? Well now we can calculate a weighted average, which is we assign a weighting to each hand to reflect its chances of being the actual hand compared to the others. There are 3 combos of AA and KK, 6 for QQ and 9 for AK. That makes a total of 3 + 3 + 6 + 9 = 21 combinations. So the chance of the hand actually being AA is 3/21 for example.

If you recall:

    Our EV against AK = 50%

                Our EV against QQ = 46%

                Our EV against KK = 34%

                Our EV against AA = 12%

So our weighted average calculation is:

50% * (9/21) +

46% * (6/21) +

34% * (3/21) +

12% * (3/21) = 41.1%

All of a sudden we have over 5% extra EV then we originally calculated, just because we “counted” our two cards.

If you are calculating EV versus a range, it is important to use combinatorics to ensure you adjust each hand based on the chances of it being the actual hand. Even if we didn’t know we had AK then it is still more likely someone has AK than AA (this should be obvious since once we get the first Ace there are only 3 other Aces left but 4 Kings). However if we have AK ourselves, it is even more unlikely our opponent has AA compared to AK (assuming we have AK then once they get their first Ace there are only 2 in the deck compared to 3 Kings).

As with a lot of my blogs, I wouldn’t worry too much about the exact maths behind this, but just try to understand the principle that our cards can make a huge amount of difference to the ranges of our opponents and also the combinations of each hand they might have. If you can get your head around this then it can often help your make the right decision in marginal spots.