What are the Odds of Flopping Two Pair In Texas Holdem?

Knowing the relative frequency at which each type of poker hand occurs in texas holdem helps gauge many different factors during play. For example, if you feel that you need your hand to improve in order to beat your opponent, you benefit by knowing how likely a hand may occur.

Today, our focus will be on the odds of flopping two pair.

Though this may seem like a simple and irrelevant statistic, once we know it we can expand upon the numbers in order to play more effectively and confidently against our opponents. But first, let's start with some raw data based on preflop holdings.

1. The odds of flopping two pair while holding unpaired cards (AKo, 87s etc.)

A simple way to calculate how often we can make two pair uses some basic probability math. First, calculate the total number of flops possible after our two-card holding is taken out: (50x49x48)/6=19600 flops. The (50x49x48) accounts for the possible 3-card flop combinations of the 50 cards left in the deck and we divide by 6 to account for redundant orders. For example:

is the same as

Then we calculate how many flops give us two pair. Say we have:

(or any unpaired hand); there are 3 of each card left in the deck for a total of 6. These can be ordered in (3x3)=9 ways and combined with the last 44 cards left in the deck (50 cards left minus the 6 cards needed for two pair) to make a flop that will give us two pair. The total number of flops that will give us two pair is thus (3x3)*(44)=396 flops. It will contain one of (3) Kings, one of (3) Queens, and one of (44) other random cards.

Take the number of flops that give us two pair and divide by the total number of flops and we get (396/19600)=2.02% of making two pair on the flop with our unpaired holding. Note that this only includes when we flop both of our cards and not times when we flop only one of our cards and the board pairs.

2. The odds of flopping two pair while holding paired cards (AA, 77 etc.)

Using the information above, this calculation will be similar and simplified. All we have to calculate is how often a paired board will flop that does not contain a card that matches our pair. Excluding our hand, there are 12 other ranks that can flop a pair and 6 ways to order them:

or

etc; to complete the flop, 44 cards are left that match neither our hand nor the flopped pair. This means there are (12x6x44)=3168 flops that will give our paired hand exactly two pair.

Taking this number and dividing it by the total number of flops and we get (3168/19600)=16.16% of flopping exactly two pair while holding a pair.

Though in isolation these probabilities may seem unimportant, they become relevant when facing opponents. After all, these probabilities apply to your opponents as well, and you can use them to determine how likely the flop helped your opponent. Indeed, these numbers can be converted into correct and calculated plays.

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