Texas Choose \"em

Introduction

Texas Choose 'Em is an easy-to-understand poker-style game developed by Dragon Fish, a provider for online casinos. The premise is straightforward: a random hand of five poker cards is displayed, and players must wager on whether another hand will be ranked higher or lower. The betting odds mirror the likelihood of a win.

Rules

1. A single 52-card deck is used.
2. Two poker hands, each containing five cards, are dealt from the same deck. The upper hand is visible while the lower hand remains concealed.
3. For each hand, a payout will be indicated. For instance, with a bet of $1, the payouts could be $1.46 for the revealed hand and $2.76 for the hidden one. All winnings are calculated on a 'for one' basis, meaning that the original bet is included in the payout.
4. Players have the option to place a bet on either hand or to abstain from betting entirely.
5. Once a player decides on which hand to wager, the hidden hand will be uncovered.
6. When a player wins, they can either reinvest their winnings into the next round or collect their earnings and start fresh.
7. If a player opts to reinvest, the previous game's concealed hand moves to the top to face a newly generated random hand.
8. It seems that a tie results in a push, although I have not personally witnessed one to confirm this.

Example

I placed a $1 bet and ended up with the hand shown. This hand is quite close to the median for five-card stud. I had to decide whether to bet on the pair of deuces, which would increase my $1 to $1.89, or on the unknown hand that would multiply to $1.90. Here's a breakdown of this scenario:

The data in the right column demonstrates that the upper hand's expected return is 94.79%, and the lower hand's is 94.71%. Even with a slightly worse return, I chose to bet on the lower hand.

The image displayed indicates that my pair of sixes from the lower hand defeated the pair of twos, allowing me to move forward.

The pair of sixes from my lower hand now occupies the top hand. I must now decide whether to place a bet on the sixes, the unknown bottom hand, or to take my $1.90 and walk away. Here’s a breakdown of this situation:

According to the right column, the top hand has an expected return of 96.08%, while the bottom hand's is 93.14%. I opted to place my bet on the sixes at the top.

The accompanying image reveals that my top pair of sixes triumphed over the ace high of the hidden hand, so I get to continue with $2.77. The ace high now shifts to the upper hand.

Next, I must choose to bet on the ace high, the concealed hand, or reclaim my $2.77. Here’s an analysis of this situation:

The right column indicates that the top hand has an expected return of 93.61%, while the bottom one offers 95.81%. I decided to place my bet on the unknown bottom hand.

The image shows that my pair of queens from the lower hand won against the ace high above, allowing me to progress with $4.35. The lower hand is now positioned on the top.

Now, I face the choice of betting on the pair of queens, the unknown lower hand, or withdrawing with my $4.35. Let’s analyze this scenario:

The right column indicates that the expected return for the upper hand is 98.01%, and for the lower hand, it’s 91.47%. I selected to bet on the queens at the top.

The image above shows my pair of queens on top prevailing over the pair of tens below, so I move ahead with $5.09. The lower hand is now on top.

For my next decision, I need to choose between the pair of tens, the unknown bottom hand, or take my $5.09 and walk away. Here’s the analysis for this situation:

The right column shows that the top hand has an expected return of 97.80%, while the bottom hand has 91.88%. I went ahead and bet on the top hand containing the pair of tens.

The image indicates that my pair of tens on top has beaten the jack high on the bottom, allowing me to go forward with $6.21. The bottom hand moves up.

Now I must decide whether to stake a bet on the jack high, the unknown lower hand, or collect my $6.21. Here’s the analysis for this situation:

The right column reveals that the top hand has an expected return of 90.87%, whereas the bottom one stands at 98.33%. I decided to gamble on the hidden lower hand.

The image above illustrates that my king high from the lower hand triumphed over the jack high from the top hand, and I continue with $6.58. The lower hand is now on top.

Next, I need to decide to bet on the king high, the unknown bottom hand, or cash out my $6.58. Here’s the analysis for this situation:

The right column indicates that the upper hand possesses an expected return of 92.20%, while the bottom one boasts 97.08%. I boldly chose to bet on the modest king high! To be honest, I felt my example was already quite illustrative and was ready to lose.

As per the image, my king high at the top outperformed the nine high from the lower hand, allowing me to move forward with $28.82! The lower hand is now relocated to the top.

For my next decision, I must choose whether to bet on the nine high, the unknown lower hand, or withdraw with my $28.82. Here’s the analysis of this situation:

The right column indicates that the top hand has an expected return of 90.23%, while the bottom has 99.28%. Despite the bottom hand offering significantly better returns, I prefer not to place that bet purely on principle. In my opinion, any risk should come with a potential reward. Therefore, I took a risk on the nine high.

Regrettably, luck was not on my side this time, and I lost to a pair of threes.

Expected

I developed a program to evaluate the likelihood of any specific hand outperforming a random hand drawn from the remaining 47 cards. The findings suggested that the odds provided by the game were reasonably aligned with the chances of winning. In reviewing roughly 40 different cases, I found no discernible advantage.

When assessing the expected value of each wager, I concluded that betting on the favored hand tends to yield the best value. The stronger the favorite, the more favorable the odds become, while the odds for the alternative hand become progressively less appealing.

Overall, a player who randomly selects which hand to bet on achieves an expected return of 94.78%, translating to a house edge of 5.22%, based on my random hand analysis. Conversely, should a player consistently choose to wager on the favorite, their expected return rises to 97.05%, resulting in a 2.95% house edge.

Return

Top Bottom Tie

Total Example Hand 5