The Probability of Forming Three of a Kind in Poker

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The likelihood of hitting Three of a Kind when you have a pocket pair stands at 10.8%, or about 1 in 9.3.

Three of a Kind refers to having three cards that share the same rank.

Example – AAAKQ

Three of a Kind can be categorized into two forms: Sets, which require a pair combined with a matching card from the community, and Trips, where a hole card is completed by a pair on the board.

While Three of a Kind Aces ranks as the top combination, kickers become vital if multiple players achieve the same hand using Trips.

Chances of Achieving Three of a Kind on the Flop

The probability of making Three of a Kind on the flop depends significantly on the preflop hand you are holding.

Probability of getting Three of a Kind from any random starting hand is 1.9%.

If you start with a pocket pair, the likelihood of hitting Three of a Kind is significantly higher at 10.8%.

For an unpaired starting hand, the odds of achieving Three of a Kind drop to 1.35%.

This illustrates that while making Three of a Kind on the flop is generally rare, having a pocket pair before the flop substantially enhances those chances.

What are the chances of flopping Three of a Kind or better?

The probability of achieving Three of a Kind or a stronger hand with a random starting hand is 2.66%.

When you hold a pocket pair, your odds of making Three of a Kind or potentially better rise to 12%.

For unpaired starting hands, the likelihood of achieving Three of a Kind or better stands at 2.08%.

Thus, not only do your chances of making Three of a Kind improve significantly with a pocket pair, but the odds of achieving any premium holdings (like Three of a Kind or better) also increase.

The probability of getting Three of a Kind remains consistent across different types of unpaired starting hands, but the chances for making Three of a Kind plus vary.

For AKo, the odds of achieving Three of a Kind are at 1.35%.

However, the odds of getting Three of a Kind or better with AKo slightly increase to 1.78%.

With T9s, the chances for forming Three of a Kind are also at 1.35%.

Yet, the probability of making Three of a Kind or better with T9s jumps to 3.58%.

All pocket pairs have a uniform frequency of achieving Three of a Kind, but higher pairs are more likely to create Top Set compared to lower or middle pairs.

Chances of Forming Three of a Kind on Later Streets

There are primarily two scenarios after the flop that can lead to making Three of a Kind:

1. Holding a pair that matches a card on the board (2 outs to achieve Three of a Kind).
2. Holding an unimproved pocket pair (also 2 outs to get Three of a Kind).

Even though the outs are the same, it's important to note the potential for extra outs when pairing with the board for two pairs.

This results in having 5 outs available to form either two pairs or trips.

From Flop to Turn

If we consider the turn, there are 2 chances out of 47 cards that will give us Three of a Kind, which translates to 2/47 = 0.0426 or approximately 4.2% .

Furthermore, there are 5 out of 47 cards that can allow us to form two pairs or better, giving us a probability of 5/47 = 0.1064 or around 10.6% .

From Turn to River

If we move to the river, the probability for hitting Three of a Kind shifts to 2 out of 46 cards, which is 2/46 = 0.0435 or roughly 4.3% .

In addition, the chances of getting two pair or better become 5 out of 46 cards, or 5/46 = 0.1087 or approximately 10.9% .

It's worth mentioning that there's a slightly higher probability of hitting a Three of a Kind when going from the turn to the river due to the reduced number of cards in play.

From Flop to River

To determine this, we can first calculate the chances of not making any hit by the river and then deduct that from 100% .

The likelihood of failing to hit Three of a Kind on the turn is 45/47.

The probability of missing Three of a Kind on the river is 44/46.

To gauge the probability of not getting two pairs or trips on the turn, we use 42/47.

Similarly, the likelihood of not achieving two pairs or trips on the river is 41/46.

Calculating the probabilities, the chance of not hitting Three of a Kind on either the turn or the river is (45/47 * 44/46) = 0.9158 or approximately 91.6% .

Furthermore, the likelihood of not hitting two pair or trips from turn to river is (42/47) * (41/46) = 0.7965 or roughly 79.7% .

Therefore -

Thus, the probability of making Three of a Kind from flop to river is approximately 8.4% .

The odds for hitting two pairs or trips from flop to river sit at around 20.3% .

Implied Odds of Three of a Kind

In Hold’em, Three of a Kind is typically regarded as a powerful hand, often strong enough to commit a full stack with 100bb effective.

However, it’s crucial to differentiate between Sets and Trips .

Both Sets and Trips are forms of Three of a Kind but arise from differing card combinations. Trips occur when one of your hole cards pairs with two community cards, while a Set is a pocket pair combined with one community card.

Regarding implied odds, Sets typically hold a higher value compared to Trips.

The underlying reason is that Sets tend to be disguised; opponents have no clear sign indicating you might have a strong hand.

In contrast, Trips are often more apparent, especially if a pair appears on the board.

For instance, if there are two Tens on the board, it becomes glaringly obvious that any player holding a Ten in their hole cards has achieved trips.

Furthermore, Sets are unique. When you form Three of a Kind with a Set, it's impossible for any other player at the table to achieve the same combination of Three of a Kind Tens due to a lack of cards.

Basic Strategy Advice

Conversely, when forming trips using board cards, there remains the possibility of opponents also holding the same Trips with higher kickers.

Typically, players will try to avoid folding Three of a Kind; however, the texture of the board should always be a consideration.

Three of a Kind shines brightest when there are no potential Straights or Flushes present on the board. In scenarios where those hands are plausible, players might need to make cautious folds with Three of a Kind when using 100bb effective.

Method (Three of a Kind)	Probability (%)
On dry board textures, it is rare to fold a Set. Exceptions might occur if an opponent seems excessively tight or when playing with a significantly larger stack (for example, stacking off with a bottom set can become questionable when stacks exceed 150bb).	1.9
Achieving Three of a Kind with a random starting hand	10.8
Hitting Three of a Kind with a pocket pair	1.35
Creating Three of a Kind with an unpaired starting hand	2.66
Getting Three of a Kind or better with a random starting hand	12
Achieving Three of a Kind or better with a pocket pair	2.08
Flopping Three of a Kind with AKo	1.35
Forming Three of a Kind or better with an unpaired starting hand	1.78
Flopping Three of a Kind with T9s	1.35
Making Three of a Kind or better with AKo	3.58
Hitting a Three of a Kind or better with T9s	4.2
Creating Three of a Kind from flop to turn while holding a pair	10.6
Getting Two Pair or better from flop to turn while holding a pair	4.3
Achieving Three of a Kind from turn to river while holding a pair	10.9
Forming Two Pair or better from turn to river while holding a pair	8.4
Understanding the Likelihood of Three of a Kind - Chance of Achieving Three of a Kind	20.3