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Pick One 21
Introduction
Pick One 21 could loosely be described as a blackjack variant. Briefly, the player bets on which of two dummy hands will come closer to 21, without going over. One card is initially dealt to each hand, to aid the player in making this decision.
Intro, second paragraph: The game made its debut in 2001 at the Sahara, where it lasted about a year. Then it disappeared for about a decade. In 2013, the game was resurrected and can be found at the WinStar World Casino in Thackerville, OK.
Rules
- The game is played with six Spanish decks. A Spanish deck is a 48-card deck, consisting of the usual cards, but excluding the four tens.
- All cards have the same point value as in blackjack, except that an ace is always one point.
- Play begins with each player making a wager.
- The dealer will then deal two cards each to a red hand and a blue hand. Each hand shall have one card face up and one face down.
- Based on the two face-up cards, the player must choose which hand to bet on. The player may bet on the red hand, blue hand or a tie. In addition, the player may double his wager if betting on the red or blue hand. The player will indicate his choice by pressing a button. If the player elects to double, then he must place the additional wager next to his original wager at this time.
- After all players have made their decisions, the dealer will turn over the face down cards. If the total points of the red hand are under 17, then the dealer will hit that hand until the total points are 17 or more. Then, the dealer will hit the blue hand in the same manner.
- The hand that comes closer to 21, without going over, shall win. As in blackjack, if one hand busts, and the other doesn't, then the non-busted hand wins. If both hands bust, then all wagers on red or blue lose. In the event of a tie, bets on red or blue shall push, except for premium match ups (see rule 10).
- A tie bet shall win if both hands have the same point value or both bust.
- A winning bet on red or blue shall pay even money and a winning bet on the tie shall pay 3 to 1.
- There are certain combinations of initial red and blue cards that strongly favor one side or the other. They are referred to as "premium match ups." If the player bets on a premium match up, and the hand results in a tie, then the bet shall lose. Premium match ups occur when one card is a 10 and the other is a 2 through a 9. A bet on the 10 in these situations is a premium match up and must beat the other hand to win; otherwise, it loses.
Strategy
The following table shows the strategy for Pick One 21, according to the first two cards.
Pick One 21 Strategy
| Red Card |
Blue Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | R/B | R | R | R | R | Rd | Rd | R | B | Bd |
| 2 | B | R/B | R | R | R | R | R | R | B | B |
| 3 | B | B | R/B | R | R | R | R | B | B | B |
| 4 | B | B | B | T | T | T | R | B | Bd | B |
| 5 | B | B | B | T | T | T | R | B | Bd | Bd |
| 6 | Bd | B | B | T | T | T | B | B | Bd | Bd |
| 7 | Bd | B | B | B | B | R | T | Bd | Bd | Bd |
| 8 | B | B | R | R | R | R | Rd | R/B | Bd | Bd |
| 9 | R | R | R | Rd | Rd | Rd | Rd | Rd | R/B | B |
| 10 | Rd | R | R | R | Rd | Rd | Rd | Rd | R | R/B |
Strategy Key
| Code | Play |
|---|---|
| R | Bet on red |
| B | Bet on blue |
| R/B | Bet on red or blue (equal probability of winning) |
| T | Bet on tie |
| Rd | Bet and double on red |
| Bd | Bet and double on blue |
Analysis
The following table shows the expected value of a non-doubled bet on Red, according to the initial two cards.
Expected Value of a Bet on Red
| Red Card |
Blue Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | -0.0876 | -0.0438 | -0.0334 | -0.0221 | -0.0087 | 0.0024 | 0.0450 | -0.0089 | -0.0766 | -0.1555 |
| 2 | -0.1565 | -0.1145 | -0.1042 | -0.0933 | -0.0811 | -0.0695 | -0.0289 | -0.0814 | -0.1474 | -0.2227 |
| 3 | -0.1770 | -0.1362 | -0.1263 | -0.1160 | -0.1038 | -0.0924 | -0.0531 | -0.1039 | -0.1660 | -0.2410 |
| 4 | -0.1997 | -0.1601 | -0.1514 | -0.1410 | -0.1286 | -0.1181 | -0.0801 | -0.1268 | -0.1885 | -0.2613 |
| 5 | -0.2247 | -0.1872 | -0.1782 | -0.1683 | -0.1567 | -0.1458 | -0.1068 | -0.1540 | -0.2140 | -0.2845 |
| 6 | -0.2476 | -0.2109 | -0.2022 | -0.1927 | -0.1813 | -0.1698 | -0.1328 | -0.1783 | -0.2360 | -0.3047 |
| 7 | -0.2175 | -0.1690 | -0.1548 | -0.1394 | -0.1226 | -0.1066 | -0.0842 | -0.1736 | -0.2326 | -0.3004 |
| 8 | -0.1502 | -0.1008 | -0.0877 | -0.0738 | -0.0570 | -0.0421 | 0.0186 | -0.0712 | -0.1746 | -0.2447 |
| 9 | -0.0712 | -0.0218 | -0.0107 | 0.0024 | 0.0180 | 0.0314 | 0.0887 | 0.0405 | -0.0622 | -0.1742 |
| 10 | 0.0196 | -0.0338 | -0.0210 | -0.0063 | 0.0108 | 0.0257 | 0.0617 | 0.0143 | -0.0492 | -0.0526 |
The following table shows the expected value of a non-doubled bet on Blue, according to the initial two cards.
Expected Value of a Bet on Blue
| Red Card |
Blue Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | -0.0876 | -0.1564 | -0.1770 | -0.1996 | -0.2249 | -0.2477 | -0.2177 | -0.1500 | -0.0711 | 0.0196 |
| 2 | -0.0438 | -0.1143 | -0.1362 | -0.1601 | -0.1870 | -0.2109 | -0.1690 | -0.1007 | -0.0218 | -0.0338 |
| 3 | -0.0335 | -0.1042 | -0.1261 | -0.1514 | -0.1782 | -0.2022 | -0.1549 | -0.0876 | -0.0107 | -0.0209 |
| 4 | -0.0222 | -0.0932 | -0.1159 | -0.1408 | -0.1684 | -0.1926 | -0.1393 | -0.0738 | 0.0023 | -0.0064 |
| 5 | -0.0089 | -0.0811 | -0.1036 | -0.1288 | -0.1564 | -0.1813 | -0.1227 | -0.0571 | 0.0179 | 0.0109 |
| 6 | 0.0024 | -0.0695 | -0.0925 | -0.1180 | -0.1458 | -0.1696 | -0.1066 | -0.0419 | 0.0316 | 0.0257 |
| 7 | 0.0447 | -0.0289 | -0.0532 | -0.0800 | -0.1070 | -0.1329 | -0.0842 | 0.0187 | 0.0887 | 0.0617 |
| 8 | -0.0089 | -0.0812 | -0.1039 | -0.1269 | -0.1541 | -0.1781 | -0.1735 | -0.0712 | 0.0404 | 0.0144 |
| 9 | -0.0765 | -0.1474 | -0.1659 | -0.1885 | -0.2140 | -0.2360 | -0.2326 | -0.1747 | -0.0623 | -0.0494 |
| 10 | -0.1554 | -0.2228 | -0.2409 | -0.2613 | -0.2845 | -0.3046 | -0.3004 | -0.2446 | -0.1743 | -0.0527 |
The following table shows the expected value of a bet on a Tie, according to the initial two cards.
Expected Value of a Bet on a Tie
| Red Card |
Blue Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 1 | -0.2527 | -0.2257 | -0.2151 | -0.2035 | -0.1910 | -0.1788 | -0.2423 | -0.2610 | -0.2813 | -0.3038 |
| 2 | -0.2255 | -0.1904 | -0.1757 | -0.1602 | -0.1424 | -0.1276 | -0.2111 | -0.2341 | -0.2553 | -0.2828 |
| 3 | -0.2152 | -0.1760 | -0.1616 | -0.1423 | -0.1231 | -0.1073 | -0.2026 | -0.2261 | -0.2515 | -0.2772 |
| 4 | -0.2032 | -0.1603 | -0.1423 | -0.1243 | -0.1028 | -0.0846 | -0.1914 | -0.2210 | -0.2438 | -0.2716 |
| 5 | -0.1911 | -0.1420 | -0.1237 | -0.1029 | -0.0815 | -0.0612 | -0.1830 | -0.2106 | -0.2366 | -0.2661 |
| 6 | -0.1790 | -0.1276 | -0.1073 | -0.0847 | -0.0613 | -0.0450 | -0.1742 | -0.2041 | -0.2311 | -0.2607 |
| 7 | -0.2425 | -0.2113 | -0.2025 | -0.1914 | -0.1827 | -0.1742 | -0.0672 | -0.2513 | -0.2817 | -0.3139 |
| 8 | -0.2609 | -0.2342 | -0.2260 | -0.2210 | -0.2105 | -0.2043 | -0.2516 | -0.1100 | -0.2936 | -0.3258 |
| 9 | -0.2809 | -0.2554 | -0.2515 | -0.2441 | -0.2366 | -0.2305 | -0.2819 | -0.2936 | -0.1502 | -0.3349 |
| 10 | -0.3038 | -0.2827 | -0.2771 | -0.2718 | -0.2660 | -0.2607 | -0.3139 | -0.3259 | -0.3350 | -0.2034 |
The next table shows the probability and return of all possible wins and losses, assuming optimal strategy. The lower-right cell shows a house edge of 2.09%.
Possible Outcomes
| Win | Probability | Return |
|---|---|---|
| 3 | 0.015734 | 0.047202 |
| 2 | 0.157029 | 0.314057 |
| 1 | 0.266599 | 0.266599 |
| 0 | 0.059950 | 0.000000 |
| -1 | 0.352623 | -0.352623 |
| -2 | 0.148066 | -0.296131 |
| Total | 1.000000 | -0.020895 |
The standard deviation is 1.41.
Following the strategy above, the house edge is 2.09%. The average bet per hand is 1.32 units, for an element of risk of 1.58%.