# Three Card Prime

## Introduction

Three Card Prime is very similar to Three Card Poker. In both games, the object is to have a higher ranking three-card poker hand the dealer. The only decision is to fold or raise after getting your hand.

## Rules

1. Player makes an Ante bet. At this time he may optionally bet on the Pair Bonus and Prime side bets.
2. The dealer gives each player three cards and himself three cards. The player may examine his own cards. The dealer's cards are dealt face down.
3. If the player made the Ante bet, then he has must either fold or raise.
4. If the player folds, then he forfeits his Ante wager.
5. If the player raises, then he must make an additional Play bet, equal exactly to his Ante bet.
6. The dealer will turn over his cards.
7. The dealer needs a queen high or better to qualify.
8. If the dealer does not qualify, then Ante will push and the Play bet will pay according to the pay table below.
9. If the dealer qualifies, then the player's hand will be compared to the dealer's hand, the higher hand wins. The order of poker hands is indicated below.
10. If the player has the higher poker hand, then the Ante will pay even money and the Play according to the pay table below.
11. If the dealer has the higher poker hand, then the Ante will lose. If the player has flush or less, the Play will also lose. In the unlikely event the player loses with a straight or higher, the Play bet will pay according to the pay table below.
12. If the player and dealer tie, then the Ante will push and the Play will pay according to the pay table below.
13. The Pair Bonus pays based mostly on the player's hand only. If the player has a paying hand of a pair or higher a dummy card will be revealed that will increase the win, if it is a deuce, according to the pay table below.
14. The Prime bet pays at least 3 to 1 if the player's three cards are the same color. If all six cards between the player and dealer hands are the same color, then it pays 4 to 1.
15. The All 6 pays based on the poker-value of the six cards comprised of the player's and dealer's three cards.

The following table shows the hand order in Three Card Prime.

### Hand Order and Probabilities in Three Card Prime

Hand Combinations Probability
Mini Royal 4 0.000181
Straight flush 44 0.001991
Three of kind 52 0.002353
Straight 720 0.032579
Flush 1,096 0.049593
Pair 3,744 0.169412
All other 16,440 0.743891

The following table shows the pay table for the Play bet. All wins on a "to one" basis.

### Play Bet Pay Table

Player
Hand
Player
Wins or Ties
Dealer
Wins
Mini royal 10 n/a
Straight flush 6 12
Three of a kind 5 10
Straight 2 4
Flush or less 1 Loss

The following table shows "pay table 1" for the Pair Bonus bet. All wins on a "to one" basis.

### Pair Bonus Pay Table 1

Player
Hand
Dummy isn't
a Deuce
Dummy is
a Deuce
Mini royal 50 75
Straight flush 40 50
Three of a kind 30 40
Straight 6 8
Flush 3 4
Pair 1 2

The following table shows "pay table 2" for the Pair Bonus, which was spotted at the Aquarius casino in Laughlin on January 20, 2020.

### Pair Bonus Pay Table 2

Player
Hand
Dummy isn't
a Deuce
Dummy is
a Deuce
Mini royal 50 75
Straight flush 40 60
Three of a kind 30 45
Straight 6 10
Flush 3 5
Pair 1 2

The Prime bet is nice and simple, as shown in the following pay table. The top award requires all player cards and dealer cards to be the same color. All pays are on a "to one" basis.

### Prime Pay Table

Event Pays
All six cards same color 4
Player cards all same color 3

The "All 6" bet pays based on the best 5-card poker hand that can be created using the three player and three dealer cards. The pay table is as follows. Wins are on a "to one" basis.

### All 6 Pay Table

Event Pays
Royal flush 1000
Straight flush 200
Four of a kind 50
Full house 25
Flush 20
Straight 10
Three of a kind 5

## Comparison to Three Card Poker

Mathematically, the base game in Three Card Prime works out to the same thing as Three Card Poker, except the player wins more if he has a straight or higher and loses to the dealer. Semantically, there are some changes too. If the dealer doesn't qualify, now it is the Ante that pushes instead of the Play. No longer is there an Ante Bonus, but the Play bet pays a premium for hands of a straight or higher. It works out to the same thing except the Play bet pays a premium for a losing hand of a straight or higher, known as "Bad Beat Bonus."

In addition, the Pair Bonus is like the Pairplus but pays a premium of a fourth dummy card is a deuce. Finally, they add another side bet called the Prime, which wins if the player's three cards are the same color.

If you're not familiar with Three Card Poker rules, read the rules section for the complete rules.

## Ante and Play Analysis

The next table shows the probability and return for all possible outcomes for the base game.

### Base Game Return Table

Event Pays Combinations Probability Return
Player loses with flush or less -2 91,126,832 0.223805 -0.447610
Player loses with straight 3 270,504 0.000664 0.001993
Player loses with three of a kind 9 3,312 0.000008 0.000073
Player loses with straight flush 11 968 0.000002 0.000026
Player ties with flush or less 1 249,216 0.000612 0.000612
Player ties with straight 2 18,288 0.000045 0.000090
Player ties with three of a kind 5 - 0.000000 0.000000
Player ties with straight flush 6 132 0.000000 0.000002
Player ties with mini royal 10 12 0.000000 0.000000
Player has flush or less, dealer doesn't qualify 1 80,937,492 0.198780 0.198780
Player has straight, dealer doesn't qualify 2 4,001,004 0.009826 0.019653
Player has three of a kind, dealer doesn't qualify 5 288,960 0.000710 0.003548
Player has straight flush, dealer doesn't qualify 6 239,316 0.000588 0.003527
Player has mini royal, dealer doesn't qualify 10 26,880 0.000066 0.000660
Player wins with flush or less 2 87,096,380 0.213906 0.427813
Player wins with straight 3 8,975,484 0.022044 0.066131
Player wins with three of a kind 6 665,776 0.001635 0.009811
Player wins with straight flush 7 570,240 0.001400 0.009803
Player wins with mini royal 11 46,804 0.000115 0.001264
Player folds -1 132,652,800 0.325792 -0.325792
Total   407,170,400 1.000000 -0.029615

The table above shows the house edge, defined as the ratio of the expected loss to the Ante bet, is 2.96%. The player will raise 67% of the time, for an average wager of 1.67. That makes the Element of Risk, defined as the expected loss to the total amount bet, 2.96%/1.67 = 1.77%.

## Pair Bonus Analysis

The following table shows the probability and return for the pay table 1 of the Pair Bonus bet. The lower right cell shows a house edge of 4.52%

### Pair Bonus Pay Table 1 Return Table

Hand Pays Combinations Probability Return
Mini royal with deuce 75 16 0.000015 0.001108
Mini royal without deuce 50 180 0.000166 0.008311
Straight flush with deuce 50 168 0.000155 0.007757
Straight flush without deuce 40 1,988 0.001836 0.073432
Three of a kind with deuce 40 204 0.000188 0.007535
Three of a kind without deuce 30 2,344 0.002165 0.064937
Straight with deuce 8 2,760 0.002549 0.020390
Straight without deuce 6 32,520 0.030030 0.180183
Flush with deuce 4 4,128 0.003812 0.015248
Flush without deuce 3 49,576 0.045781 0.137342
Pair with deuce 2 14,112 0.013032 0.026063
Pair without deuce 1 169,344 0.156380 0.156380
All other -1 805,560 0.743891 -0.743891
Total   1,082,900 1.000000 -0.045205

The following table shows the probability and return for the pay table 2 of the Pair Bonus bet. The lower right cell shows a house edge of 3.38%

### Pair Bonus Pay Table 2 Return Table

Hand Pays Combinations Probability Return
Mini royal with deuce 75 16 0.000015 0.001108
Mini royal without deuce 50 180 0.000166 0.008311
Straight flush with deuce 60 168 0.000155 0.009308
Straight flush without deuce 40 1,988 0.001836 0.073432
Three of a kind with deuce 45 204 0.000188 0.008477
Three of a kind without deuce 30 2,344 0.002165 0.064937
Straight with deuce 10 2,760 0.002549 0.025487
Straight without deuce 6 32,520 0.030030 0.180183
Flush with deuce 5 4,128 0.003812 0.019060
Flush without deuce 3 49,576 0.045781 0.137342
Pair with deuce 2 14,112 0.013032 0.026063
Pair without deuce 1 169,344 0.156380 0.156380
All other -1 805,560 0.743891 -0.743891
Total   1,082,900 1.000000 -0.033802

## Prime Analysis

The following table shows the probability and return for the Pair Bonus bet. The lower right cell shows a house edge of 3.62%

### Prime Return Table

Hand Pays Combinations Probability Return
Six-card prime 4 9,209,200 0.022618 0.090470
Three-card prime 3 86,595,600 0.212677 0.638030
Loser -1 311,365,600 0.764706 -0.764706
Total   407,170,400 1.000000 -0.036206

## All 6

The following table shows my analysis of the All 6 bet. The combinations were taken from my Probabilities in Poker page. The lower right cell shows a house edge of 10.22%.

### All 6 Return Table

Hand Pays Combinations Probability Return
Royal flush 1000 188 0.000009 0.009234
Straight flush 200 1,656 0.000081 0.016268
Four of a kind 50 14,664 0.000720 0.036014
Full house 25 165,984 0.008153 0.203826
Flush 20 205,792 0.010108 0.202168
Straight 10 361,620 0.017763 0.177626
Three of a kind 5 732,160 0.035963 0.179817
All other -1 18,876,456 0.927202 -0.927202
Total   20,358,520 1.000000 -0.102248