Does a 3 of a Kind Beat 3 Pairs in Poker?

As a passionate poker player and content creator, this is a question I get a lot from newer players. The short answer is yes, a 3 of a kind (also called a set or trips) will beat 3 pairs in standard poker games like Texas Hold‘em. But why is that? Let‘s dig deeper into the math and probabilities behind different poker hands.

Poker Hand Rankings and Odds

First, a quick refresher on general poker hand rankings and odds of being dealt different hands, from rarest to most common:

As you can see, the odds of getting a 3 of a kind is 1 in 46, while getting two pair is significantly more common at 1 in 20.

So right away based on the math, we know that trips is rarer and thus ranks higher than two pair in the standard hand rankings, which are based on statistical improbability.

Analyzing Trips vs Three Pair Hands

Now let‘s analyze some example trips vs three pair hands:

• 3 of a Kind: K♥ K♠ K♦ 7♣ 2♠
• 3 Pairs: A♠ A♥ 9♦ 9♣ 5♠ 5♦

Even though the three pair hand has higher pairs of aces and nines, the three of a kind is still the statistically rarer hand. Any set, no matter how low it is, will beat any two or three pair combination.

Let‘s examine why:

Probability of Getting Trips

• There are 13 ranks of cards that trips can be made of
• For each rank, the probability is:
  • (3/51) (2/50) (1/49) = 1 in 20,400
• Since there are 13 ranks, the total odds become:
  • 13 * (1/20,400) = 1 in 1,600
• For any trips:
  • 1 – (51/52) (50/51) (49/50) = 1 in 46

So trips can form in any of 13 ranks, leading to odds of around 1 in 46.

Probability of Getting Three Pairs

• There are 13 ranks for the first pair
• For each remaining rank, odds of another pair is:
  • (2/50) * (1/49) = 1 in 625
• For the 3rd pair it‘s the same math
• In total for three pairs:
  • 13 (1/625) (1/625) = 1 in 482,200

As you can see, the odds of getting three pair is astronomically harder than getting a set. This extreme improbability makes any trips hand superior to three pairs.

When Can Three Pairs Win?

The only time three pairs would ever beat trips is in a specifically designed game variation like 3-5 Triple Draw (winning low hand is best). Most standard poker games go by the classic rankings and odds we discussed above.

So in Texas Hold‘em, Omaha, 7-Card Stud and other popular games, even the highest three pairs will always lose to the simplest three of a kind. There are no exceptions to trips beating any two or three pair hands.

Final Thoughts

While new players often think having more pairs must be better, the statistical improbability of making certain hands is what determines rankings and winners in poker. I hope this analysis has clearly laid out why a 3 of a kind beats 3 pairs every time. Let me know if you have any other poker math questions!