Analyzing Odds in Two-Player Texas Hold 'Em

Introduction

This section takes a close look at the odds for the final hand of a hypothetical player, labeled player two, based on the hand value of the opposing player, termed player one. The combinations provided are calculated from a total of combin(52,5)×combin(47,2)×combin(45,2) = 2,781,381,002,400. The main goal of this section is to help understand the probabilities of bad beats in a two-person game scenario, such as the situation presented in the Bad Beat Bonus. Ultimate Texas Hold \"Em .

To illustrate, if you're curious about the likelihood of a specific player achieving a full house only to lose against a four of a kind, we can refer to table 7, which reveals that there are 966,835,584 combinations for this scenario. Furthermore, this table indicates that if player one holds a full house, the chance of losing to a four of a kind stands at 0.013390. To determine the overall probability before any cards are dealt, you would divide 966,835,584 by the total number of possible hands, 2,781,381,002,400, which results in a probability of 0.0002403.

Table 1 details the various combinations available for each hand of the second player, in situations where the first player holds nothing better than a high card.

Table 2 presents the combinations possible for each of the second player's hands when the first player has a single pair.

Table 3 provides the combinations for each potential hand of the second player, based on the first player having two pairs.

Table 4 outlines the combinations possible for each hand of the second player if the first player has three of a kind.

Table 5 showcases the combinations available for the second player given that the first player has achieved a straight.

Table 6 reveals the combinations for each hand of the second player in the event that the first player has a flush.

Table 7 lists the combinations available for each of the second player's hands when the first player holds a full house.

Table 8 describes the number of combinations for each hand of the second player when the first player has four of a kind.

Table 9 demonstrates the combinations for each hand of the second player when the first player has a straight flush.

Table 10 indicates the combinations for each hand of the second player based on the first player achieving a royal flush.

The subsequent table provides an overview of the combinations for each hand of player one, categorized by the hand that ultimately won.

The next table examines the probabilities related to each hand of player one depending on who the winning player is. The last line shows that both players have a 47.97% chance of emerging victorious, along with a 4.07% possibility of a tie.

Written by: Michael Shackleford