Does 3 of a Kind Beat 2 Pair in Poker

Introduction

The rank of poker hands determines which hand will win in a showdown. While some rankings might seem intuitive, others require a bit more thought. One of the most common questions that arises is: does 2 Pair beat 3 of a Kind?

The answer is no - 3 of a Kind is a stronger hand than 2 Pair.

Let’s explore the reasoning behind Poker hierarchy and why 3 of a Kind holds the upper hand.

What is 3 of a Kind?

Three of a Kind, also referred to as ‘trips’ or a set’, is a hand where you have three cards of the same rank and two unrelated cards. For example, Q♠ Q♥ Q♦ 7♠ 3♣ is a Three of a Kind in Queens.

There are two common ways to make 3 of a Kind:

• Set: When you have a pocket pair (like Q♠ Q♦) and one matching card appears on the board (like Q♥).
• Trips: When you have one card in your hand that matches two cards on the board (like having Q♠ when the board shows Q♥ Q♦).

What is 2 Pair?

Two Pair is a hand where you have two different pairs of cards, plus one unrelated card. For example, 8♠ 8♣ 4♦ 4♠ K♠ is a 2 Pair with Eights and Fours.

To make 2 Pair, you will need:

• Two cards of the same rank and another two cards of a different rank.
• A fifth card that is unrelated to either pair, acting as a kicker.

Two Pair is a strong hand, but it is more common than Three of a Kind, which is part of the reason it ranks lower.

Why Does 3 of a Kind Beat 2 Pair

Here are the reasons why 3 of a King is ranked higher than 2 Pair.

1. Probability

Probability of 2 Pair:

Here’s how to calculate the probability of 2 Pair:

Choosing Ranks for the Pairs: There are 13 possible ranks (2 through Ace), and we need to choose two ranks for the pairs. The number of ways to choose two ranks out of 13 is calculated using combinations:

(13, 2) = 78

Choosing Suits for Each Pair: For each chosen rank, there are four suits, and we need to pick two suits. The number of ways to choose two suits out of four is:

(4, 2) = 6

Since we have two pairs, we multiply this by itself for each pair:

6 × 6 = 36

Choosing the Fifth Card: For the fifth card, we can choose from the 11 remaining ranks (since two ranks are already used). We need one card from these ranks, and there are four suits for that card:

11 × 4 = 44

Calculating Total Possibilities for 2 Pair: Multiply the number of ways to choose the ranks, suits for pairs, and the fifth card:

78 × 36 × 44 = 123,552

The total number of 5-card poker hands is:

(52, 5)=2,598,960

So, the probability of 2 Pair is:

123,552/2,598,960 ≈ 4.75%

Probability of 3 of a Kind:

Let's calculate the probability:

Choosing the Rank for 3 of a Kind: There are 13 possible ranks, and we need to choose one for the Three of a Kind:

(13, 1)=13

Choosing Suits for 3 Cards of the Same Rank: From the four suits, we choose three suits for the Three of a Kind:

(4, 3) = 4

Choosing Two Other Cards: For each of the two remaining cards, we select a rank from the 12 remaining ranks, with four suit choices for each card. Since the order of these two cards does not matter, we divide by 2 to avoid double-counting:

(12, 2) × 4 × 4/2 = 66 × 16 = 528

Calculating Total Possibilities for 3 of a Kind: Multiply the ways to choose the ranks and suits:

13 × 4 × 528 = 27,072

Probability of 3 of a Kind:

27,0722/598,960 ≈ 1.39%

Since Three of a Kind (1.39%) is less likely to occur than Two Pair (4.75%), it ranks higher in poker.

2. Hand Combination

Hand Combinations for 2 Pair

In a 2 Pair hand, we have:

• Two ranks that form the pairs.
• One rank that makes up the fifth card.

The possible combinations for a 2 Pair hand are determined by selecting two ranks to form pairs, which can be done in 78 ways. Then, each of these ranks can be paired up in 36 different ways (each pair has (4, 2) = 6 combinations of suits. The fifth card can be any of the 44 remaining options (11 ranks left, with 4 suit choices each).

So, as calculated earlier:

Total Combinations for 2 Pair = 123,552

Hand Combinations for 3 of a Kind

In a 3 of a Kind hand, we have:

• One rank for the three cards.
• Two other different ranks for the remaining two cards.

The possible combinations for Three of a Kind are determined by choosing one rank for the three matching cards, which can be done in 13 ways. Each Three of a Kind has 4 ways to form with suits, and the two other cards can be chosen from the remaining 12 ranks and then suited in 528 different ways.

As calculated earlier:

Total Combinations for Three of a Kind=27,072

Rarity of Hand Combinations

Comparing the numbers:

• Two Pair has 123,552 possible combinations.
• Three of a Kind has 27,072 possible combinations.

With fewer possible ways to form Three of a Kind than Two Pair, it is statistically less likely to occur. This rarity contributes to its higher rank, as poker rankings prioritize hands that are harder to achieve.

Frequently Asked Questions

Can 2 Pair ever beat 3 of a Kind?

No, in standard poker hand rankings, 2 Pair cannot beat 3 of a Kind. However, the strength of a 2 Pair hand can sometimes be deceptive, especially if a player holding 3 of a Kind does not reveal their hand until the end. While Two Pair may feel like a strong hand, 3 of a Kind remains statistically superior.

Why is 3 of a Kind rarer than 2 Pair?

Three of a Kind is rarer because the specific combination of three identical cards requires either two of those cards in your hand or two of them appearing on the board. In contrast, 2 Pair only requires matching two different pairs, which occurs more frequently given the 52-card deck.

How to play 2 Pair against a potential 3 of a Kind?

If the board shows pairs or tripling cards that match your opponent’s style of play, proceed cautiously. Watching for raises and betting patterns can give clues to your opponent’s strength. Remember, 3 of a Kind remains statistically superior, so unless you are sure your 2 Pair is the best hand, avoid committing too many chips.

Conclusion

Three of a Kind beats Two Pair because it is less common and offers more room for enhancement. Poker hand rankings are structured to reward hands that are more challenging to achieve and have greater potential for dominance. Although 2 Pair is a respectable hand, 3 of a Kind surpasses it as a stronger option, providing an edge in any Poker showdown.