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Last Updated: May 14, 2015

# Texas Choose 'em

## Introduction

Texas Choose 'Em is a simple poker-based game by Dragon Fish, who supplies games to Internet casinos. The idea is simple — a random five-card poker hand is shown. The player may bet whether another poker hand will be higher or lower than the hand shown. The odds for each bet are commensurate with the probability of winning.

## Rules

1. A single 52-card deck is used.
2. Two 5-card poker hands are dealt from the same deck. The top hand is face up and the bottom hand is face down.
3. A payoff will be shown for each hand. For example, based on a \$1 bet, the payoffs might be \$1.46 on the top hand and \$2.76 on the bottom one. All wins are on a "for one" basis, meaning the payoff includes the return of the original wager.
4. The player may choose to bet on either hand or neither.
5. After choosing a hand to bet on, the bottom hand will be revealed.
6. If the player wins, he may parlay his winnings on the next hand or "collect" his win and start over.
7. If the player chooses to parlay, then the bottom hand from last game will be moved to the top to compete with a new random hand.
8. I assume that a tie is a push, although I've never seen one occur to verify that.

## Example

I bet \$1 and got the hand above. This happens to be close to the median five-card stud hand. My choices were to bet on the pair of deuces and multiply my \$1.00 to \$1.89 or on the unknown hand and multiply to \$1.90. Here is an analysis of this situation:

### Example Hand 1

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 1.89 1.89 769,282 0.501508 0.947867
Bottom 1.90 1.90 764,630 0.498475 0.947102
Tie 27 0.000018
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 94.79% and the bottom one 94.71%. Despite the lower return, I bet on the bottom hand.

The image above shows that my pair of sixes on the bottom hand beat the pair of twos, so I got to advance.

The pair of sixes in the bottom hand is moved to the top hand. My choice is to bet on that, the unknown bottom hand, or quit and collect my \$1.90. Here is an analysis of this situation:

### Example Hand 2

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 2.77 1.457895 1,010,892 0.659017 0.960811
Bottom 5.19 2.731579 523,020 0.340965 0.931374
Tie 27 0.000018
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 96.08% and the bottom one 93.14%. I chose to bet on the pair of sixes on top.

The image above shows that my pair of sixes on top beat the ace high below, so I get to advance with \$2.77. The ace high from the bottom is moved to the top hand.

My next choice is to bet on the ace high, the unknown bottom hand, or quit and collect my \$2.77. Here is an analysis of this situation:

### Example Hand 3

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 6.65 2.400722 597,847 0.389746 0.936109
Bottom 4.35 1.570397 935,850 0.610096 0.958093
Tie 242 0.000158
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 93.61% and the bottom one 95.81%. I chose to bet on the unknown bottom hand.

The image above shows that my pair of queens on bottom beat the ace high on top, so I get to advance with \$4.35. The bottom hand is moved to the top hand.

My next choice is to bet on the pair of queens, the unknown bottom hand, or quit and collect my \$4.35. Here is an analysis of this situation:

### Example Hand 4

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 5.09 1.170115 1,284,691 0.837511 0.980061
Bottom 24.49 5.629885 249,221 0.162471 0.914695
Tie 27 0.000018
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 98.01% and the bottom one 91.47%. I chose to bet on the pair of queens on top.

The image above shows that my pair of queens on top beat the pair of tens on bottom, so I get to advance with \$5.09. The bottom hand is moved to the top hand.

My next choice is to bet on the pair of tens, the unknown bottom hand, or quit and collect my \$5.09. Here is an analysis of this situation:

### Example Hand 5

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 6.21 1.220039 1,229,568 0.801576 0.978043
Bottom 23.57 4.630648 304,344 0.198407 0.918752
Tie 27 0.000018
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 97.80% and the bottom one 91.88%. I chose to bet on the pair of tens on top.

The image above shows that my pair of tens on top beat the jack high on bottom, so I get to advance with \$6.21. The bottom hand is moved to the top hand.

My next choice is to bet on the jack high, the unknown bottom hand, or quit and collect my \$6.21. Here is an analysis of this situation:

### Example Hand 6

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 78.43 12.629630 110,242 0.071869 0.908657
Bottom 6.58 1.059581 1,423,454 0.927973 0.983263
Tie 243 0.000158
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 90.87% and the bottom one 98.33%. I chose to bet on the unknown bottom hand.

The image above shows that my king high on bottom beat the jack high on top, so I get to advance with \$6.58. The bottom hand is moved to the top hand.

My next choice is to bet on the king high, the unknown bottom hand, or quit and collect my \$6.58. Here is an analysis of this situation:

### Example Hand 7

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 28.82 4.379939 322,537 0.210267 0.921995
Bottom 8.09 1.229483 1,211,160 0.789575 0.970769
Tie 242 0.000158
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 92.20% and the bottom one 97.08%. I chose to throw caution to the wind and bet on the meager king high! To be honest, I thought my example had long since made its point and wanted to lose.

The image above shows that my king high on top beat the nine high on bottom, so I get to advance with \$28.82! The bottom hand is moved to the top hand.

My next choice is to bet on the nine high, the unknown bottom hand, or quit and collect my \$28.82. Here is an analysis of this situation:

### Example Hand 8

Hand Win Pays
(for one)
Combinations Probability Expected
Return
Top 3656.11 126.860167 10,856 0.007077 0.902343
Bottom 28.82 1.000000 1,522,842 0.992766 0.992766
Tie 241 0.000157
Total 1,533,939 1.000000

The right column shows the top hand has an expected return of 90.23% and the bottom one 99.28%. Although the bottom hand has the much greater return, note that the bet can only win or push. I refuse to bet that on principle alone. Any risk should have some kind of reward. So I throw a Hail Mary on the nine high.

Unfortunately, Mary wasn't on my side on that one and I lost to a pair of threes.

## Analysis

I wrote a program to analyze the probability of any given hand beating a random hand from the other 47 cards. The results were that the odds offered by the game were fairly commensurate with the probability of winning. In looking at about 40 examples, the number of times I detected an advantage was zero.

In calculating the expected value of each bet, I find that the better value is to bet on the favorite. The greater the favorite, the better the odds, and the worse the odds on the other hand.

Overall, the player picking which hand to bet on randomly has an expected return of 94.78%, or a house edge of 5.22%, based on my sample of random hands. If the player always bets on the favorite, then I show his expected return is 97.05%, or a house edge of 2.95%.

## Video

I created a video of the example hands above.

Written by: Michael Shackleford

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