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21 to the River
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Introduction
21 to the River is a blackjack side bet I noticed at the Hard Rock in Las Vegas on March 27, 2008. The rules are as follows.
- Player makes a blackjack and poker bet.
- The blackjack bet shall be adjudicated according to conventional blackjack rules. In the case of the Hard Rock, these rules were six decks, dealer hits soft 17, double after split allowed, no surrender, and no re-splitting aces.
- In the event the blackjack hand busts, a five-card poker hand will be created using the card that busted the player, and the next four cards in the shoe.
- The poker hand shall pay according to the return table below.
- If the player does not bust, then the poker bet will push.
- If the player splits, the first hand, if any, that busts will start the poker hand. If none bust, the poker bet will push.
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The first step to analyze this game is to determine the probability that the blackjack hand will bust, and if so, with what card. To answer this, I ran the following simulation. My simulation treats all 10-point cards the same way, so I divided that total between the four 10-point cards.
Blackjack Events
| Event | Probability |
|---|---|
| Player busts with 6 | 0.003197 |
| Player busts with 7 | 0.006978 |
| Player busts with 8 | 0.012119 |
| Player busts with 9 | 0.017238 |
| Player busts with 10 | 0.023341 |
| Player busts with J | 0.023341 |
| Player busts with Q | 0.023341 |
| Player busts with K | 0.023341 |
| No bust | 0.867104 |
| Total | 1.000000 |
The total probability of busting comes to 13.29%. This is higher than other places on my site, which say it is 13.00%. This is because of the splitting rule.
The next table shows the probability of each poker hand, with six decks, according to the first card dealt in the hand.
The next table combines the probability of each bust card, by the poker probabilities starting with that card.
21 to the River — Return Table
| Event | Pays | Probability | Return |
|---|---|---|---|
| Royal flush | 500 | 0 | 0.000158 |
| Straight flush | 250 | 0.000001 | 0.000365 |
| Five of a kind | 100 | 0.000003 | 0.000308 |
| Four of a kind | 50 | 0.000222 | 0.011081 |
| Full house | 25 | 0.000485 | 0.012136 |
| Flush | 15 | 0.000467 | 0.007010 |
| Straight | 10 | 0.000453 | 0.004533 |
| Three of a kind | 5 | 0.005558 | 0.027792 |
| Two pair | 2 | 0.008712 | 0.017425 |
| Jacks or better | 0 | 0.023288 | 0.000000 |
| No bust | 0 | 0.867104 | 0.000000 |
| All other | -1 | 0.093704 | -0.093704 |
| Total | 1.000000 | -0.012895 |
The lower right cell shows a house edge of 1.29%, per bet made. The probability of the bet resolving with a win or loss is 10.96%. The house edge, per bet resolved, is thus 1.29%/10.96% = 11.76%.
Buster Blackjack
Buster Blackjack is a side bet I noticed at the Sycuan casino, near San Diego, on November 30, 2008. The bet wins if the dealer busts, the more cards it takes, the more the player wins. The following table shows the probabilities and return for a six-deck game, where the dealer hits a soft 17. The lower right cell shows a house edge of 6.21%.
Buster Blackjack — Six Decks, Dealer Hits Soft 17
| Event | Pays | Probability | Return |
|---|---|---|---|
| Bust with 8+ cards | 250 | 0.000012 | 0.002986 |
| Bust with 7 cards | 50 | 0.000214 | 0.010722 |
| Bust with 6 cards | 12 | 0.002638 | 0.031651 |
| Bust with 5 cards | 4 | 0.020473 | 0.08189 |
| Bust with 4 cards | 2 | 0.089392 | 0.178784 |
| Bust with 3 cards | 2 | 0.173032 | 0.346064 |
| Dealer doesn't bust | -1 | 0.714241 | -0.714241 |
| Total | 1.000000 | -0.062143 |
The next table shows the return, according to the number of decks, and whether the dealer hits or stands on a soft 17.
Buster Blackjack — Expected Return
| Decks | Stand Soft 17 | Hit Soft 17 |
|---|---|---|
| 1 | -0.087690 | -0.068890 |
| 2 | -0.084766 | -0.065097 |
| 4 | -0.083066 | -0.062915 |
| 5 | -0.082707 | -0.062455 |
| 6 | -0.082462 | -0.062143 |
| 8 | -0.082153 | -0.061749 |
As long as I went to the trouble to analyze this bet, the next table shows a finer breakdown of the possible dealer outcomes in a six-deck game, and the dealer hits a soft 17.
Possible Dealer Outcomes — Six Decks, Dealer Hits Soft 17
| Event | Probability |
|---|---|
| Total of 17 | 0.133459 |
| Total of 18 | 0.141205 |
| Total of 19 | 0.135682 |
| Total of 20 | 0.181531 |
| Total of 21 | 0.0748744 |
| Blackjack | 0.0474895 |
| Bust with 3 cards | 0.173032 |
| Bust with 4 cards | 0.0893918 |
| Bust with 5 cards | 0.0204726 |
| Bust with 6 cards | 0.0026376 |
| Bust with 7 cards | 0.000214444 |
| Bust with 8 cards | 0.000011528 |
| Bust with 9 cards | 0.00000040805 |
| Bust with 10 cards | 0.00000000909509 |
| Bust with 11 cards | 0.00000000011986 |
| Bust with 12 cards | 0.000000000000824386 |
| Bust with 13 cards | 0.00000000000000222834 |
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