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- Appendices
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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.
- 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.
21+3 — Version 3 — 3-8 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,
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'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.
21+3 — Version 6
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%.
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The breakeven point, with no house edge, is at a jackpot of $10,957.37.
Written by: Michael Shackleford
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