I must begin with a disclaimer: This post is directed towards poker novices and/or non-poker playing family and friends. No smarmy comments from poker pros who might be reading this. 🙂

A fairly common question from the general public to a poker professional is, ‘do you count cards?’ While the phrase ‘counting cards’ usually refers to black jack and keeping a running total of the high / low cards remaining in the shoe, there is somewhat of an analogy in poker. Counting combinations of hands your opponent can possibly hold is an essential part of good hand reading. I am going to begin with a quick overview of combinatorics and then give a simple poker hand example.

Counting Combinations

How many different ways can we be dealt pocket aces?* Let’s write them all out:

1)

2)

3)

4)

5)

6)

As you can see, there are a total of 6 different ways you can be dealt AA. The same is true with any other pocket pair 22-KK. If we do the same thing for two cards of different rank (such as AK), we find that we can be dealt each of those hands 16 different ways, 12 of those being unsuited and 4 being suited.

When we are dealt two cards randomly from a 52 card deck, it turns out there are 1326 total combinations of hands we can be dealt (52*51/2). We can use this fact to determine the probability of being dealt a certain hand or type of hand preflop:

AA: 6 / 1326 = 0.45% chance of being dealt pocket aces
22-AA: (13*6) / 1326 = 5.9% chance of being dealt a pocket pair
AK: 16 / 1326 = 1.2% chance of being dealt ace-king
AT-AK: (4*16) / 1326 = 4.8% chance of being dealt an ace with ten kicker or better

Card-Removal Effects

If we start preflop with a hand like KQ, we know an opponent can possibly have 6 combinations of AA in his starting hand. However, if we start with AK, our opponent can only make 3 different combinations of AA. By holding a single ace in our hand, the probability our opponent has AA is cut in half. If we then go on to see a flop of A72, our opponent can only hold a single combination of AA in his hand range. As you can see, as a hand progresses and more information becomes known, we can use card-removal effects to narrow down our opponent’s hand range. This is essentially how a poker professional ‘counts cards’.

Poker Hand Example

Let’s assume we are playing an opponent who is very passive and only raises when he has the nuts (the best possible hand) or the near-nuts. He also does not slowplay strong hands.

We raise preflop with AA and our opponent calls.

Flop: JT2r (r means ‘rainbow’, i.e. all cards are different suits). We bet and our opponent calls.
Turn: 6r, we bet and our opponent calls.
River: A, we bet and our opponent raises.

On the river we have the 2nd best possible hand (three of a kind aces) and are only losing to a single hand that our opponent can hold: king-queen which makes a straight. We resist the urge to instantly reraise our opponent and instead pause to consider his hand range. Since our opponent doesn’t slowplay, we can eliminate other three of a kind type hands from his range (JJ, TT, 22, and 66). The river ace has to have improved his hand to the near nuts, since he raised.

Let’s assume his hand range is AJ, AT, and KQ. If we were not proficient at counting combos, we might say we are usually winning since we beat AJ and AT and only lose to KQ. With no card removal, AJ and AT are each 16 combos and KQ is 16 combos. However, in this specific example there are some very strong card-removal effects. We hold 2 aces in our hand, there is an ace on the river, and a J and T on the flop. This reduced our opponent’s combinations of AJ and AT to 3 each, so we are beating a total of 6 combos and losing to 16. What at first might appear to be a great river card becomes pretty terrible once our passive opponent puts in a raise.

So there you have it… a basic intro into counting cards in poker. Please post comments or questions and let me know if you are interested in more posts like this. Thanks for reading!

*For a mathematical discussion on combinatorics, please visit http://www.mathsisfun.com/combinatorics/combinations-permutations.html