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Three Card Mulligan Poker
Introduction
Introduction
Three Card 2nd Chance is a poker-based game, invented and patented by yours truly, and marketed by Shuffle Entertainment. The thrust of the game is that if the player doesn't like his first hand, he may exchange it for a second hand by merely increasing his bet. The dealer also has the ability to replace his hand in certain circumstances. Then, the higher hand wins, and all wins pay at least even money.
Rules
- A single 52-card deck is used. All hands are scored according to conventional poker rules, except the hand order is altered for 3-card hands.
- Play starts with the player making an Ante bet. The player may optionally make an optional Pair Plus bet.
- The dealer shall give the player three cards and himself three cards. Player cards will be face up and dealer cards face down.
- The player has the choice to either stand or to trade in his hand for a new three-card hand, known as taking a "Mulligan."
- If the player chooses to take a Mulligan, then he should discard his cards and make a Raise wager equal to the Ante wager. The player may also make a new Pair Plus wager on the Mulligan hand.
- The dealer will then turn over his cards.
- If the dealer has a king high or higher, he will stand.
- If the dealer has a queen high or less, he will discard his hand for three new cards.
- The dealer will then compare her own hand to the player's hand, and the higher hand wins. If the dealer has the higher hand, then the player will lose his Ante wager as well as any Raise wager. If the player has the higher hand, then both Ante wager and Raise wager will be paid according to the Ante and Raise pay table below. This table also reflects the hand ranking order. A tie will result in a push.
Ante and Raise Pay Table
| Hand | Pays |
|---|---|
| Straight flush | 6 to 1 |
| Three of a kind | 4 to 1 |
| Straight | 3 to 2 |
| All other | 1 to 1 |
Strategy
The player should take keep a hand of king-ten or higher, otherwise take a Mulligan. In other words, the lowest hand the player should keep it king-ten-2, and the highest hand he should take a Mulligan on is king-9-8.
Ante Only Analysis
The next table shows the possible outcomes if the player bets the Ante only, switching with K87 or less.
Ante Only Return Table — Switch with K87 or Less
| Event | Bet | Pays | Probability | Return |
|---|---|---|---|---|
| Straight flush | 2 | 12 | 0.001075 | 0.012894 |
| Three of a kind | 2 | 8 | 0.001160 | 0.009284 |
| Straight flush | 1 | 6 | 0.002169 | 0.013014 |
| Three of a kind | 1 | 4 | 0.002341 | 0.009365 |
| Straight | 2 | 3 | 0.015675 | 0.047025 |
| Less than straight | 2 | 2 | 0.169427 | 0.338855 |
| Straight | 1 | 1.5 | 0.031575 | 0.047363 |
| Less than straight | 1 | 1 | 0.286811 | 0.286811 |
| Tie | 1 | 0 | 0.000608 | 0.000000 |
| Tie | 2 | 0 | 0.000500 | 0.000000 |
| Loss | 1 | -1 | 0.176960 | -0.176960 |
| Loss | 2 | -2 | 0.311698 | -0.623396 |
| Total | 1.000000 | -0.035744 |
The lower right cell shows a house edge of 3.57%. The player will raise 49.95% of the time, for an average final wager of 1.4995 units. That makes theElement of Risk 0.035744/1.4995 = 2.38%.
Pair Plus
The Pair Plus wager on the initial hand has the same pay table and odds as in Three Card Poker. The following table shows the exact odds. The lower right cell shows a house edge of 7.28%.
Pair Plus — Original Hand
| Event | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Straight flush | 40 | 48 | 0.002172 | 0.086878 |
| Three of a kind | 30 | 52 | 0.002353 | 0.070588 |
| Straight | 6 | 720 | 0.032579 | 0.195475 |
| Flush | 3 | 1096 | 0.049593 | 0.148778 |
| Pair | 1 | 3744 | 0.169412 | 0.169412 |
| Loser | -1 | 16440 | 0.743891 | -0.743891 |
| Total | 22100 | 1.000000 | -0.072760 |
If the player remakes the Pair Plus bet on the Mulligan hand the odds change, assuming the player follows the optimal strategy of taking a Mulligan on king-ten or less. The lower right cell of the following table shows the house edge for the Pair Plus on the Mulligan hand is 7.65%, higher by 0.38% compared to the initial hand.
Pair Plus — Mulligan Hand
| Event | Pays | Probability | Return |
|---|---|---|---|
| Straight flush | 40 | 0.002155 | 0.086192 |
| Three of a kind | 30 | 0.002335 | 0.070036 |
| Straight | 6 | 0.032363 | 0.194179 |
| Flush | 3 | 0.049547 | 0.148641 |
| Pair | 1 | 0.169018 | 0.169018 |
| Loser | -1 | 0.744582 | -0.744582 |
| Total | 1.000000 | -0.076515 |
Methodology
This analysis was done by random simulation, except the initial Pair Plus wager. It was assumed that the dealer shuffles after every hand.
With the possibility of both player and dealer switching cards, the number of possible combinations is combin(52,3) × combin(49,3) × combin(46,3) × combin(43,3) = 76,277,828,779,152,000, which would take years to loop through.