Poker Strategy - 7/8 - Combinations and Card Removal

This is the seventh part of an eight-part guide to the basics of poker strategy. If you have already read this tutorial, then you can access the eighth part here. If you missed the previous part, "Ranges", then you'll find it here.

Combinations and Card Removal

You're playing a hand of Texas Hold'em poker and you've narrowed down that your opponent holds either a pair of queens, or an ace and a king. The question you need to ask yourself is - which is the more likely hand for your opponent to have?

You may consider that any combination of two hole cards is as likely as another other combination, but that's an incorrect assumption to make, and I'm going to show you why:

The possible hole cards for a pair of queens are Q♠Q♥ Q♠Q♦ Q♠Q♣ Q♥Q♦ Q♥Q♣ and Q♦Q♣ - a total of six hands.

The possible hole cards for an ace and a king are A♠K♠ A♠K♥ A♠K♦ A♠K♣ A♥K♠ A♥K♥ A♥K♦ A♥K♣ A♦K♠ A♦K♥ A♦K♦ A♦K♣ A♣K♠ A♣K♥ A♣K♦ and A♣K♣, which is a total of sixteen hands.

So out of a possible (six + sixteen) twenty-two hands, six out of twenty-two (around 27 percent) are a pair of queens, whilst sixteen out of twenty-two (around 73 percent) are an ace and a king. This means that it is three times more likely that your opponent has an ace and a king rather than a pair of queens.

When you are doing your poker analysis, you need to understand the odds of the different card combinations appearing. You'll also need to add the hash (#) symbol to your knowledge of poker notation. A hash symbol means "the number of total combinations" - so "QQ AK (#22)" means "there are a total of 22 combinations within this range".

You don't need to perform these mental gymnastics when you're calculating the number of combinations. Instead, you just need to remember the following:

Card Removal

Often there are known cards that need to be taken into consideration when you're working out the odds of certain combinations. There are the cards in your hand, and any community cards that have been dealt.

For example, say you're looking at a flop of Q♥9♣6♠. You're making the assumption by the way play has progressed so far that your opponent has an open-ended straight-draw or a set. Therefore, the complete range is QQ, 99, 66 and JT - how many combinations is that? You already know that pairs have six combinations, so that's eighteen as there are three possible sets. JT can either be off-suit or suited, so that's another sixteen combinations. The grand total is therefore thirty-four - and as there are eighteen sets and "only" sixteen open-ended straight-draws, that makes a set more likely than a open-ended straight-draw, right?

Wrong! Your opponent cannot have a possible six combinations of each pair in their hand. For example, if he has QQ then there are only three possible combinations he could have as the Q♥ is already on the board, so he can only have Q♠Q♦, Q♠Q♣ or Q♦Q♣. The same goes for each possible set - meaning there are only nine possible sets he could have, instead of eighteen. Suddenly the odds of an open-ended straight-draw are much higher - up to sixty-four percent, as opposed to thirty-six percent for a set.

You also need to factor in your hole cards when using card removal. Say for example in our sample hand you hold Q♣T♠ - you can withdraw another two cards from the possible QQ sets (as the only remaining possiblity is Q♠Q♦), lowering the odds of a set right down to twenty-eight percent.

Knowing how to calculate card combinations and using card removal is an important tool in understanding ranges. Once you have worked out your opponents' range, you can then work out the odds of each possible hand in that range and come to a conclusion about the hand he is most likely to have, based on accurate probabilities. From that point you can then work out how likely it is that your hand is going to beat them.

This concludes the seventh lesson in an eight-part series on the basics of poker strategy. To view the next article, click here.