Last Updated: September 10, 2010

# Three Card Blackjack

## Introduction

Three Card Blackjack is an easy-to-play blackjack variant found in many Washington state casinos, especially the smaller ones. It made its Las Vegas debut on January 1, 2010, at the Monte Carlo.

## Rules

1. A single 52-card deck is used.
2. Player makes an Ante wager and an optional Ace Plus side bet.
3. Dealer gives the player and himself three cards. One dealer card is dealt face up, the other two face down. Player cards are dealt face down, which the player may look at.
4. Hands will be scored according to the best blackjack hand that can be composed using any two, or all three, cards.
5. After examining his cards, the player may fold or raise.
6. If the player folds, he forfeits his Ante wager, but the Ace Plus wager, if made, will remain in play.
7. If the player decides to raise, then he must make a Raise wager equal to his Ante wager.
8. The dealer will turn over his two face-down cards.
9. The dealer must have at least 17 points to open.
10. If the dealer cannot open, and the player has a blackjack, then the player will be paid 1 to 1 on the Ante and the Raise will push.
11. If the dealer cannot open, and the player does not have a blackjack, then the Ante and Raise will push.
12. If the dealer can open, and the player has a blackjack, then the Ante and Raise shall pay 1 to 1, even if the dealer also has a blackjack.
13. If the dealer can open, and the player does not have a blackjack, then the higher hand shall win. If the player has the higher hand, then the Ante and Raise shall pay even money. If the dealer has the higher hand, then the Ante and Raise shall lose. If there is a tie, then the Ante and Raise shall push.
14. The Ace Plus side bet pays according only to the player's hand. The possible pay tables are listed towards the bottom of this page.

## Ante Odds

The first table shows the probability of each number of points for the player or dealer hand.

### Points Distribution

Points Combinations Probability
6 4 0.000181
7 24 0.001086
8 48 0.002172
9 92 0.004163
10 136 0.006154
11 200 0.009050
12 268 0.012127
13 356 0.016109
14 536 0.024253
15 788 0.035656
16 1,156 0.052308
17 2,016 0.091222
18 2,680 0.121267
19 3,336 0.150950
20 6,304 0.285249
21 1,532 0.069321
BJ 2,624 0.118733
Total 22,100 1.000000

The next table shows the probability of each possible outcome on the Ante bet. The lower right cell shows a house edge of 3.42%. The player will raise 63.66% of the time, so the average player wager will be 1.6366 units. Thus the element of risk, the ratio of expected player loss to the average wager, is 2.09%.

### Ante Return Table

Event Pays Combinations Probability Return
Player wins 2 341,489,680 0.279563 0.559127
Dealer doesn't open and player has blackjack 1 25,599,648 0.020957 0.020957
Push 0 257,354,144 0.210685 0.000000
Player folds -1 443,840,928 0.363354 -0.363354
Dealer wins -2 153,226,800 0.125440 -0.250881
Total   1,221,511,200 1.000000 -0.034151

The next table shows the expected value according to the player's hand and the dealer's up card. The player should fold on an expected value of less than -1. Cells have been conveniently colored red when the odds favor folding, and green for raising.

### Expected Return According to Player Total and Dealer Up CardExpand

Player 2 3 4 5 6 7 8 9 10 A
6 -1.198582 -1.400709 -1.570922 -1.691489 -1.647163 -1.750000 -1.835106 -1.900709 -1.964539 -1.939716
7 -1.198582 -1.393617 -1.542553 -1.684397 -1.640071 -1.744681 -1.828014 -1.893617 -1.957447 -1.934397
8 -1.195745 -1.372340 -1.523303 -1.677305 -1.632979 -1.738475 -1.820922 -1.887411 -1.951241 -1.930851
9 -1.186613 -1.358447 -1.512691 -1.670213 -1.626349 -1.732347 -1.814292 -1.881129 -1.947117 -1.925686
10 -1.172716 -1.346336 -1.501082 -1.663475 -1.624086 -1.727576 -1.808824 -1.877451 -1.943680 -1.921464
11 -1.160938 -1.333718 -1.491671 -1.657144 -1.622263 -1.723569 -1.805745 -1.875177 -1.941064 -1.918085
12 -1.150270 -1.323772 -1.482011 -1.650887 -1.617612 -1.720871 -1.803016 -1.872658 -1.938340 -1.915423
13 -1.137891 -1.311755 -1.472290 -1.643099 -1.613256 -1.717045 -1.799437 -1.868804 -1.936011 -1.913337
14 -1.127030 -1.304801 -1.467954 -1.637029 -1.606136 -1.712907 -1.796782 -1.867458 -1.935575 -1.914244
15 -1.117876 -1.296842 -1.458041 -1.630912 -1.601026 -1.708667 -1.793447 -1.864429 -1.933488 -1.912748
16 -1.098594 -1.274028 -1.439573 -1.618587 -1.595245 -1.700223 -1.784653 -1.857419 -1.928355 -1.907273
17 -0.902922 -1.073912 -1.226339 -1.268578 -1.326859 -1.321114 -1.611843 -1.711354 -1.819351 -1.816612
18 -0.559604 -0.713710 -0.732127 -0.591323 -0.734567 -0.644870 -0.884957 -1.402702 -1.546080 -1.575111
19 -0.238346 -0.251290 -0.140474 0.046776 -0.080634 -0.017875 0.023597 -0.468811 -1.139250 -1.198903
20 0.264071 0.406365 0.535628 0.719856 0.642774 0.713253 0.782869 0.788370 0.116521 -0.657262
21 0.733564 0.902661 1.055977 1.234992 1.198258 1.297247 1.369686 1.450641 1.204872 -0.361228
BJ 1.511082 1.601961 1.687655 1.780186 1.771266 1.828191 1.876197 1.917465 1.957863 1.960106

## Ace Plus Odds

The next table shows the return for the most common pay table for the Ace Plus side bet, known as "pay table 2." An A represents an ace, a T any 10-point card, and x a 2-9 card. The lower right cell shows a house edge of 2.75%, which for a side bet isn't bad.

### Ante Return Table for Pay Table 2

Event Pays Combinations Probability Return
AAA 100 4 0.000181 0.018100
AAT 25 96 0.004344 0.108597
AAx 15 192 0.008688 0.130317
ATT 6 480 0.021719 0.130317
ATx 3 2,048 0.092670 0.278009
Axx 1 1,984 0.089774 0.089774
No aces -1 17,296 0.782624 -0.782624
Total   22,100 1.000000 -0.027511

The next table shows all six pay tables available from the game maker and the house edge for each.

### Ante Return Table

Hand P.T. 1 P.T. 2 P.T. 3 P.T. 4 P.T. 5 P.T. 6
AAA 100 100 100 100 100 100
AAT 20 25 30 25 40 25
AAx 10 15 12 10 25 10
ATT 5 6 6 7 6 6
Atx 3 3 3 3 1 3
Axx 2 1 1 1 1 1
House Edge 2.46% 2.75% 3.19% 4.92% 6.08% 7.10%

## Strategy

As shown in the table above, the player should raise according to his best hand as follows:

• 16 or less: Never
• 17: Dealer 2
• 18: Dealer 2-8
• 19: Dealer 2-9
• 20-BJ: Always

For purposes of comparing one bet to the other, I recommend using the element of risk. For the Ace Plus side bet, the element of risk is the same as the house edge. The element of risk on the Ante bet is 2.09%, which is lower than any pay table on the Ace Plus. So, if you want to lose as little as possible, then I recommend betting the Ante only.

Be warned, if you don't bet something on the Ace Plus, it never fails that the dealer and other players will falsely tell you that the side bets are "where the money is." If you don't bet the Ace Plus, and get three aces, I can guarantee you that you'll really get lectured. With this game, and any casino game, follow such advice at your own peril.

## Methodology

This analysis was based on a combinatorial program in C++. Thus, the results can be considered exactly correct.

## Acknowledgements

I would like to thank gaming mathematician Eliot Jacobson for answering my many questions about the game and confirming and analysis.

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