# 21 to the River

## 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.

1. Player makes a blackjack and poker bet.
2. 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.
3. 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.
4. The poker hand shall pay according to the return table below.
5. If the player does not bust, then the poker bet will push.
6. 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.

### Six-Deck Poker Probabilities by First CardExpand

Poker Hand A up 2 up 3 up 4 up 5 up 6 up 7 up 8 up 9 up 10 up J up Q up K up
Five of a kind 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023 0.000023
Royal flush 0.000003 0 0 0 0 0 0 0 0 0.000003 0.000003 0.000003 0.000003
Straight flush 0.000003 0.000007 0.00001 0.000014 0.000017 0.000017 0.000017 0.000017 0.000017 0.000014 0.00001 0.000007 0.000003
Four of a kind 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668 0.001668
Full house 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653 0.003653
Flush 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516 0.003516
Straight 0.001729 0.001729 0.002593 0.003458 0.004322 0.004322 0.004322 0.004322 0.004322 0.004322 0.003458 0.002593 0.001729
Three of a kind 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826 0.041826
Two pair 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558 0.065558
Jacks or better 0.25072 0.091171 0.091171 0.091171 0.091171 0.091171 0.091171 0.091171 0.091171 0.091171 0.25072 0.25072 0.25072
All other 0.6313 0.790849 0.789982 0.789114 0.788246 0.788246 0.788246 0.788246 0.788246 0.788246 0.629565 0.630432 0.6313
Total 1 1 1 1 1 1 1 1 1 1 1 1 1

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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