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Card Removal - Decreasing Number of Holdem Hands

25 March 2009, By: Pokerjunkie.com
When you put opponents on hand ranges and try to measure your hand's chances against those, basic hand frequencies need to be combined with the concept of card removal.
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We talked about the basic hand frequencies in Texas Holdem and how you use them when you put an opponent on a range.

Now let's look closer at what happens when cards are removed, that is, when they're not available anymore, for some reason.

Reasons for card removal

A card can be removed from the available cards for three reasons:

  • You have it in your hand
  • It appears on the board
  • It's in an opponent's hand

The last point is basically only valid for games like seven card stud, where you get to see some of the opponents' cards. Or, of course, in holdem if a card is inadvertently displayed by a player. Or the dealer, which is not that unusual.

Adjusting your calculations

As cards are revealed during the hand, you need to constantly update your calculation (or estimation) of your hand's chances in the pot.

Firstly, of course, each action at the table causes you to modify your picture of the opponent's likely holdings. This is a subjective job, based on experience and intuition.

Secondly, though, the objective grounds for your analysis change as well. Before you can even start using your intuition and experience, you need to update the hard facts that they are based on.

As cards are revealed, the likelihoods of the opponent's possible holdings change accordingly. We'll look at the numbers and suggest a simplified method to keep track of them at the table.

Card removal for pairs

The number of combinations for pair hands is 6. This is the basic frequency for those hands. When cards are removed, the number of combinations go down rapidly:

Removed cards Combinations for each pair
0 6
1 3
2 1
3 0
4 0

Card removals for non-pairs

Non-pair hands come in 16 variations in an untouched deck. As cards are removed, the number of combinations goes down. But the exact numbers depend on what exact cards are gone.

For example, AK comes in 16 combinations and if one ace is removed, 12 combinations remain. But when a second card is removed, the remaining combinations depend on whether it's a second ace or an ace and a king. If two aces are gone, eight combinations remain. If it's an ace and a king, there are nine combinations left.

With three cards removed, there are either six or four combinations left. With three cards gone, there can be four, three or zero combinations left. Zero if all four aces or all four kings are gone. And so on.

Removed cards Combinations
0 16
1 12
2 9, 8           (9)
3 6, 4           (5)
4 4, 3, 0       (4)
5 2, 0           (2)
6 1, 0           (0)

The numbers in the parentheses are the average, or most likely value, that we suggest you use in your estimations. After all, you'll not be 100% accurate no matter what you do. Approximations and simplifications are fine, or even necessary.

Some examples

Below are a few examples of how you can use the numbers above. We've used the average or approximate number, and provide the exact number in parentheses.

  • You have AK. The number of AK for the opponent is 9.
  • You have AA. The number of AK for the opponent is 9 (actually 8).
  • You have AK. The number of AA for the opponent is 3.
  • You have AA. The number of AA for the opponent is 1.
  • You have AK, there's an A on board. The number of AK for the opponent is 5 (actually 6).
  • You have AK, there's an A on board. The number of AA for the opponent is 1.
  • You have AA, there's an A on board. The number of AK for the opponent is 5 (actually 4).
  • You have AA, there's an A on board. The number of AA for the opponent is 0.

Summing up the numbers

As usual, you need to sum up all the numbers and combine them with your hand's chances against each and every one of the possible hands that your opponent might hold.

In a later article, we'll show a smart way of summing up all this information to reach a decision. After all, at the poker table, only the decisions that we make are interesting. The rest is just accidental occurrences.

/Charlie River

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