﻿ black 29 rouletteSix Card Poker - ugslot of Odds

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Last Updated: July 17, 2013

# Six Card Poker

## Rules

1. Six Card Poker is a poker variant game played heads up against the dealer. The object of the game is to win by having a better five-card poker hand than the dealer using six cards.
2. The game is played using one standard 52-card deck. The game uses standard poker rules for scoring and comparing hands.
3. To begin, the player makes an Ante Bet.
4. The dealer deals six cards face down to the player. The dealer deals to himself, three cards face up and three cards face down.
5. The player examines his cards and must either fold the hand, losing his Ante, or raise by making an additional wager that is equal to his Ante. Players may not share information about their cards with other players.
6. The dealer reveals his hole cards and compares his best five-card hand to the player’s best five-card hand. The dealer must qualify with ace-king or higher.
7. If the dealer does not qualify, then the player’s Ante bet is a push. The raise bet is resolved as follows:
• If the dealer's hand beats the player's hand, then the player loses his raise bet.
• If the player's hand beats the dealer's hand, then the player is paid 1 to 1 on his raise bet.
• In the event of a tie, the player’s raise bet is a push.
8. If the dealer does qualify:
• If the dealer's hand beats the player’s hand, then the player loses his Ante and raise bets.
• If the player's hand beats the dealer’s hand, then the player is paid 1 to 1 on his Ante and raise bets.
• In the event of a tie, the player’s Ante and raise bets push.
9. There are also two optional side bets, the Aces Up and Two-Way Bad Beat, which are explained below.

## Analysis

The game was analyzed through brute force, combinatorial analysis. The following table summarizes the analysis results. The bottom left cell shows the house edge of 1.2717%. With an average bet size of 1.6987, the element of risk is 0.7486%.

### Ante Bet Analysis

Outcome Dealer Qualifies Net Win Combinations Probability Return
Player Wins yes 2 1,040,023,036,938,350 0.272693 0.545387
Player Wins no 1 755,514,209,224,548 0.198095 0.198095
Ties yes 0 210,847,746,240 0.000055 0.000000
Ties no 0 50,992,983,876 0.000013 0.000000
Fold - -1 1,149,186,512,820,950 0.301316 -0.301316
Dealer Wins no -1 2,937,744,249,924 0.000770 -0.000770
Dealer Wins yes -2 865,968,968,971,212 0.227056 -0.454113
Total     3,813,892,312,935,100 1.000000 -0.012717

## Strategy

Two basic playing strategies are presented below.

Intermediate basic strategy:
1. Fold if dealer has a higher ranked hand using his three up cards alone. This rule supersedes all other rules.
2. Fold with ace-10 or lower.
3. Raise with ace-king or higher.
4. With ace-queen, fold if the dealer is showing an ace or king.
5. With ace-jack, fold if the dealer is showing an ace, king, or queen.

Simple basic strategy:
1. Fold if dealer has a higher ranked hand using his three up cards alone. This rule supersedes all other rules.
2. Raise with ace-king. Fold otherwise.

• With optimal play, the house edge is 1.27%.
• Using the intermediate basic strategy, the house edge is 1.48 %.
• Using the simple basic strategy, the house edge is 1.63%.

## Aces Up

Aces Up is a side bet based on the poker-value of the player's hand only. The following table shows what each hand pays, the probability, and contribution to the return. The lower right cell reflects a house edge of 3.30%.

### Aces Up Analysis

Event Pays Combinations Probability Return
Royal flush 500 188 0.000009 0.004617
Straight flush 100 1,656 0.000081 0.008134
Four of a kind 30 14,664 0.000720 0.021609
Full house 10 165,984 0.008153 0.081530
Flush 8 205,792 0.010108 0.080867
Straight 6 361,620 0.017763 0.106576
Three of a kind 4 732,160 0.035963 0.143853
Two pair 2 2,532,816 0.124411 0.248821
Aces 1 751,332 0.036905 0.036905
Loser -1 15,592,308 0.765886 -0.765886
Total   20,358,520 1.000000 -0.032973

The "Two-Way Bad Beat" pays if either you or the dealer has at least a pair of aces and loses. The higher the losing hand, the more the bet pays. In other words, it pays based on the worse hand between you and the dealer, as long as they don't tie.

The following table shows all the odds. The zeros for the probability and return of a losing straight flush are not a mistake. The probability of that is about 1 in 25 trillion. Such a situation was the basis of the movie Honeymoon in Vegas, but I'm getting off topic. The lower right cell of the table below shows a house edge of 10.84%.

Event Pays Probability Return
Straight Flush 10000 0.000000 0.000000
Four of a Kind 5000 0.000001 0.006250
Full House 500 0.000099 0.049370
Flush 200 0.000307 0.061440
Straight 100 0.000972 0.097168
Three of a Kind 35 0.004056 0.141965
Two Pair 10 0.034513 0.345127
Pair of Aces 9 0.015036 0.135328
Loser -1 0.945016 -0.945016
Total   1.000000 -0.108367

## Acknowledgments

Usually, I like to do my own math. However, in this case , who is marketing the game, kindly provided the math report by mathematician Cindy Liu, which the Ante bet analysis is based on. The Two Way Bad Beat is based on the work of Elliot Frome. I did the analysis of Aces Up.

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