Last Updated: July 11, 2019

# 21 + 3

## Version 1

Version 1 of 21+3 I noticed at the Las Vegas Hilton in April, 2001. The side bet pays based on the player's first two cards and the dealer's up card. If the three cards equal a flush, straight, straight flush, or three of a kind the side bet pays 9 to 1. The following table shows the probability of each hand in a six-deck game,as played at the Hilton.

### 21+3 — 6 decks

Hand Combinations Probability Pays Return
Straight flush 10368 0.002068 9 to 1 0.018613
Three of a kind 26312 0.005248 9 to 1 0.047236
Straight 155520 0.031021 9 to 1 0.279192
Flush 236736 0.047221 9 to 1 0.424993
Pair+flush 56160 0.011202 9 to 1 0.100819
Pair (no flush) 977184 0.194918 Loss -0.194918
Nothing 3551040 0.708321 Loss -0.708321
Total 5013320 1   -0.032386

Following is the house edge for 4, 6, and 8 decks.

• 1 deck — 13.30%
• 2 decks — 7.26%
• 4 decks — 4.24%
• 6 decks — 3.24%
• 8 decks — 2.74%

## Version 2

At the Regent (now known as the Rampart) in Las Vegas all hands listed above, plus a pair, pay 5 to 2. I'll call this version 2. Two decks are used in this version. The following table shows a house edge under these rules of 2.78%.

### 21+3 — 2 decks

Hand Combinations Probability Pays Return
Straight flush 384 0.002109 2.5 to 1 0.005272
Three of a kind 728 0.003998 2.5 to 1 0.009994
Straight 5760 0.03163 2.5 to 1 0.079076
Flush 8768 0.048148 2.5 to 1 0.120371
Pair 34944 0.19189 2.5 to 1 0.479726
Nothing 131520 0.722225 Loss -0.722225
Total 182104 1   -0.027786

## Version 3

I have an unconfirmed report that Internet casinos using Wagerworks software use the following pay table, which I will call "Version 3."

### 21+3 — Version 3 — Six Decks

Hand Pays Combinations Probability Return
Suited three of a kind 100 1040 0.000207 0.020745
Three of a kind 33 25272 0.005041 0.166352
Straight flush 35 10368 0.002068 0.072383
Straight 10 155520 0.031021 0.310214
Flush 5 292896 0.058424 0.292118
Loss -1 4528224 0.903239 -0.903239
Total   5013320 1 -0.041427

Although Wager Works only uses six decks in their blackjack game, as far as I know, here is the house edge for 3 to 8 decks.

Decks House Edge
3 7.76%
4 5.99%
5 4.89%
6 4.14%
7 3.60%
8 3.18%

## Version 4

This version, with a 30-20-10-5 pay table, is known as "21+3 Xtreme." The following return table shows the house edge for six decks is 13.39% (ouch!).

### 21+3 — Version 4 — Six Decks

Hand Pays Combinations Probability Return
Straight flush 30 10,368 0.002068 0.062043
Three of a kind 20 26,312 0.005248 0.104968
Straight 10 155,520 0.031021 0.310214
Flush 5 292,896 0.058424 0.292118
Loser -1 4,528,224 0.903239 -0.903239
Total   5,013,320 1.000000 -0.133896

If you happen to see it with other than six decks, here is the expected value for various number of decks.

### 21+3 — Version 4 -- 1 to 8 Deck Summary

Decks Expected
Value
1 -0.227330
2 -0.172736
4 -0.143780
5 -0.137863
6 -0.133896
8 -0.128912

The Caesars Entertainment Blog mentions that the xtreme version of 21+3 can be found in the Party Pit at the Paris casino in Las Vegas. I had to chuckle at their sense of humor with this remark,

"We would calculate the odds of hitting these side bets, but being in Vegas isn't about math, it's about having fun. And just never-you-mind that this is the blog of a casino company. Check out the new Ooh La La Party Pit at Paris Las Vegas the next time you’re in the neighborhood.

Hey, it is about the math everywhere, even in the Party Pit!

## Version 5

I saw this version at an Internet casino using live dealers from Evolution Gaming. Following is the return table, based on eight decks.

### 21+3 — Version 5 — 8 Decks

Event Pays Combinations Probability Return
Suited three of a kind 100 2,912 0.000244 0.024446
Straight flush 40 24,576 0.002063 0.082524
Three of a kind 25 61,568 0.005169 0.129213
Straight 10 368,640 0.030947 0.309465
Flush 5 700,928 0.058841 0.294207
Loser -1 10,753,536 0.902736 -0.902736
Total   11,912,160 1.000000 -0.062882

Following is the house edge for this pay table for various numbers of decks.

• 4 decks: 8.78%
• 5 decks: 7.81%
• 6 decks: 7.14%
• 7 decks: 6.29%

## Version 6

I have an unconfirmed report that the Ameristar in Kansas City is using the following pay table with two decks.

### 21+3 — Version 6 — 2 Decks

Event Pays Combinations Probability Return
Straight flush 7 384 0.002109 0.014761
Three of a kind 7 728 0.003998 0.027984
Straight 7 5,760 0.031630 0.221412
Flush 7 10,016 0.055002 0.385011
Pair 0 33,696 0.185037 0.000000
Loser -1 131,520 0.722225 -0.722225
Total   182,104 1.000000 -0.073057

The next table shows the house edge of the Version 6 pay table by number of decks.

Decks House Edge
1 13.70%
2 7.31%
4 4.15%
6 3.10%
8 2.58%

## Version 7

This pay table I hear is used by live dealer casinos by Evolution Gaming, which use eight decks.

### 21+3 — Version 7 — 8 Decks

Event Pays Combinations Probability Return
Suited three of a kind 100 2,912 0.000244 0.024446
Straight flush 40 24,576 0.002063 0.082524
Three of a kind 30 61,568 0.005169 0.155055
Straight 10 368,640 0.030947 0.309465
Flush 5 700,928 0.058841 0.294207
Loser -1 10,753,536 0.902736 -0.902736
Total   11,912,160 1.000000 -0.037039

Here is the house edge for four and six decks.

• Four decks: 6.39%
• Six decks: 3.70%

## Version 8

I think I saw this version at the Global Gaming Expo, but I'm not sure. This version is noteworthy because it pays a progressive jackpot for three suited aces. Here is the full pay table. Pays are on a "to one" basis.

### Version 8 Pay Table

Event Pays
Suited three aces Jackpot
Suited three of a kind 125
Straight flush 25
Three of a kind 20
Straight 7
Flush 3

The following table shows my analysis based on a six-deck game.

### Version 8 Return Table — Six Decks

Event Pays Combinations Probability Return
Suited three aces ? 80 0.000016 ?
Suited three of a kind 125 960 0.000191 0.023936
Straight flush 25 10,368 0.002068 0.051702
Three of a kind 20 25,272 0.005041 0.100819
Straight 7 155,520 0.031021 0.217150
Flush 3 292,896 0.058424 0.175271
Loser -1 4,528,224 0.903239 -0.903239
Total   5,013,320 1.000000 -0.334360

The lower right cell shows that a house edge of 33.44%, before considering the value of the jackpot. If the minimum bet is \$1, then each \$1,000 in the meter decreases the house edge by 1.60%. If the minimum bet is \$5, then each \$10,000 in the meter decreases the house edge by 3.19%.

The breakeven points, where the house edge is zero, is \$27,016.90 for a \$1 bet and \$135,084.50 for a \$5 bet, based on six decks.

## Version 9

I saw this version at the Rampart casino in Las Vegas on January 29, 2021 . This version is similar to version 8, except a suited three of a kind in queens to aces is required to hit the jackpot and it is a \$1 "red light" bet, meaning pays are on a "for one" basis. Following is the pay table.

### Version 9 Pay Table

Event Pays
Suited three of a kind (Q's-A's) Jackpot
Suited three of a kind (2's-J's) \$125
Straight flush \$25
Three of a kind \$20
Straight \$6
Flush \$2

The following table shows my analysis based on a six-deck game.

### Version 9 Return Table — Six Decks

Event Pays Combinations Probability Return
Suited three of a kind (Q's-A's) Jackpot 240 0.000048 ?
Suited three of a kind (2's-J's) \$125.00 800 0.000160 0.019947
Straight flush \$25.00 10,368 0.002068 0.051702
Three of a kind \$20.00 25,272 0.005041 0.100819
Straight \$6.00 155,520 0.031021 0.186128
Flush \$2.00 292,896 0.058424 0.116847
Loser \$- 4,528,224 0.903239 0.000000
Total   5,013,320 1.000000 0.475444 + ?

The lower right cell an expected return of 47.54%, before considering the value of the jackpot. Each \$1,000 in the jackpot increases the return by 4.79%. For example, at the time I saw this side bet the jackpot was at \$3,886.15, for an expected return of 66.15%.

The breakeven point, with no house edge, is at a jackpot of \$10,957.37.

## Version 10

Concept Gaming, a provider of games for Internet casinos, offers this bet, calling it just the "poker" bet. They use six decks and a 100-35-20-15-5 pay table. Wins are on a "for one" basis. The following table shows a return to player of 95.14%. The help file claims a return of 95.11%. You may choose whom to believe.

### Version 10 Return Table — Six Decks

Event Pays Combinations Probability Return
Suited three of a kind 100 1,040 0.000207 0.020745
Straight flush 35 10,368 0.002068 0.072383
Three of a kind 20 25,272 0.005041 0.100819
Straight 15 155,520 0.031021 0.465320
Flush 5 292,896 0.058424 0.292118
Loser 0 4,528,224 0.903239 0.000000
Total   5,013,320 1.000000 0.951386

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