Understanding the Odds of Flopping Set Over Set in Texas Holdem

When it comes to Texas Holdem, one of the most thrilling moments for a player is hitting a set. But what are the odds of hitting a set over set, i.e., when both players have pocket pairs and hit a set on the flop? This article delves into the probabilities and calculations involved in this scenario.

Setting the Stage

Let's assume two players, Player A and Player B, both start the game with pocket pairs. Player A has a pocket pair of 7s, i.e., 77, and Player B has a pocket pair of 9s, i.e., 99. The flop consists of three community cards that will be dealt on the table.

Calculating the Probabilities

Probability of Both Players Having Pocket Pairs

The probability that Player A is dealt a pocket pair is approximately 1/221 because there are 13 different ranks, and for each rank, there are 6 combinations of two cards (since there are 13 ranks, and each rank has 2 out of the 4 cards in the deck). For Player B, the probability of being dealt a pocket pair is similarly 1/221, but this is conditional on Player A having a pair.

Flopping a Set

The next step is to calculate the probability of flopping a set. For Player A, to flop a set, one of the three flop cards must match their pocket pair. There are 50 unseen cards after dealing Player A's pocket pair, and 2 of those cards are 7s. The probability that none of the flop cards are a 7 can be calculated as follows:

Probability that none of the flop cards are a 7 48/50 * 47/49 * 46/48.

Therefore, the probability that at least one of the flop cards is a 7 (flop a set) is:

1 - (48/50 * 47/49 * 46/48) ≈ 0.117

The same calculation applies for Player B flopping a set with their pocket pair of 9s, which is also approximately 0.117.

Final Calculation

To find the odds of both players flopping sets, we need to consider the combined probability:

Combined Probability: Since both events are independent, the total probability of both players flopping sets is:

PSet A * PSet B ≈ (1/221) * (1/221) * 0.117 * 0.117.

This calculation gives approximately:

≈ (0.01369 / 2212) ≈ (0.01369 / 48841) ≈ 0.00000028.

Conclusion

The odds of flopping set over set in Texas Holdem, given both players start with pocket pairs, are extremely low, around 0.000028 or roughly 1 in 36,000. Keep in mind that this is a rough estimate and can vary based on the specific pocket pairs and the community cards.

This discussion provides a detailed look at the probabilities involved in this thrilling scenario, offering a better understanding of the game dynamics and improving your strategic game plan.

Note: All probabilities are calculated under standard Texas Holdem rules with a 52-card deck and proper community card distribution.