Blackjack side bet

Hi again,
There is a side bet where you can bet that the dealer will get an "8 or more card 21".
Side bet and game info
Cost $1 to play side bet
Pays: $1,000,000
6-deck
1 What is the chance of the dealer getting an 8 or more card 21 when:
(a) the dealer stands soft 17
(b) the dealer hits soft 17
2. What are the chances when the dealer's up card is an:
(a) Ace, (b) two, and (c) three.

For "1(a)" I think the chance is about 1 in 1.08 million, but I am not sure.

Thanks for your time

Quote: dddkkk1

Hi again,
There is a side bet where you can bet that the dealer will get an "8 or more card 21".
Side bet and game info
Cost $1 to play side bet
Pays: $1,000,000
6-deck
1 What is the chance of the dealer getting an 8 or more card 21 when:
(a) the dealer stands soft 17
(b) the dealer hits soft 17
2. What are the chances when the dealer's up card is an:
(a) Ace, (b) two, and (c) three.

For "1(a)" I think the chance is about 1 in 1.08 million, but I am not sure.

Thanks for your time

dddkkk1,

I ran four 1-billion-round CVData sims to check this sidebet. Below are the number of times it hit overall, as well as the number specifically for A-up, 2-up, 3-up, and (anything else)-up. Note that, if the number of hits were exactly 1,000, the bet would be EV-neutral.

Decks	H/S 17	Total	A-up	2-up	3-up	other	EV	SD	N0
8	S	1,081	232	429	254	166	8.1%	1,040	164,761,621
8	H	1,636	510	588	307	231	63.6%	1,279	4,044,541
6	S	1,051	202	459	212	178	5.1%	1,025	404,075,739
6	H	1,573	511	555	309	198	57.3%	1,254	4,790,928

As you can see, someone erred, as the SB is +EV for each configuration. As expected, the EV is higher for 8D than for 6D, and for H17 rather than S17. However, the SD is huge, so you'll need to bring a BIG bankroll ;-)

Hope this helps!

Dog Hand

P.S. A tip of the hat to miplet for his table-maker!

Would this be countable if many high cards have left the building?

There’s probably a max table payout in the fine print.

Yes. Very dependent on aces and deuces (and fives?) with reasonably high EORs.

Easiest way to solve should be with a Markov chain

Quote: dddkkk1

Hi again,
There is a side bet where you can bet that the dealer will get an "8 or more card 21".
Side bet and game info
Cost $1 to play side bet
Pays: $1,000,000
6-deck
1 What is the chance of the dealer getting an 8 or more card 21 when:
(a) the dealer stands soft 17
(b) the dealer hits soft 17
2. What are the chances when the dealer's up card is an:
(a) Ace, (b) two, and (c) three.

For "1(a)" I think the chance is about 1 in 1.08 million, but I am not sure.

Thanks for your time

I’ve had one 8 card 20 in my life followed by 4 blackjacks in a row crazy day that was

Quote: heatmap

I’ve had one 8 card 20 in my life followed by 4 blackjacks in a row crazy day that was

That was just the guy in the back room who rigs the shuffle machines messing with you

I have created a spreadsheet with all of the 8-card 21s listed for a H17 game.

For 8 decks, I calculate that the probability of an 8-card 21 is 1.498 E-06. So, that's a 49.8% edge for the player, but of course its on a bet with lottery-type odds.

For 6 decks, I calculate that the probability of an 8-card 21 is 1.42755 E-06. A 42.755% player edge.

Given the tax bite out a $1M payout, there is some question whether there is any real-life player advantage.

Edit: To be clear, these numbers are only for 8-card 21s; 9-11 card 21s are not included in the above numbers.

Here is a table of Effect of Removal (EOR) in %.

Card	8 Decks EOR,%	6 Decks EOR,%
T	2.925	3.742
9	2.826	3.621
8	2.588	3.323
7	2.035	2.616
6	0.817	1.047
5	-1.513	-1.955
4	1.108	1.386
3	-1.430	-1.885
2	-7.260	-9.292
A	-10.869	-13.826

These values are quite remarkable, aren't they? Certainly the largest EOR values I have ever calculated!

In the 6 deck game, if a single Ace is removed/dealt, the Player edge decreases by about 13.8%, from 42.755% to about 29.8%. If 4 Aces are removed, the sidebet becomes unfavorable: the House has an edge of 6.2%.

In the 6 deck game, if a single Ten is removed/dealt, the Player edge increases by about 3.74%. If 4 Tens are removed, the player edge rises from 42.755% to 58.403%. That's an increase of 15.64% so even accounting for the smaller number of cards remaining the EOR for 4 Tens is a bit more then 4X the EOR for 1 Ten. That is, the EOR for multiple cards is more than linear.

This is a mathematically extreme bet that I thought math nerds would find interesting. I feel its okay to post this info because I don't believe this is a credible AP opportunity due to the tax bite, the enormous bankroll and time required to grind out this side-bet, as well as the possibility of large losses from playing the wagers on the main BJ game for a million hands or so.

Quote: michael99000

That was just the guy in the back room who rigs the shuffle machines messing with you

the last blackjack was a push with the dealer so maybe ;-)

Quote: DogHand

...

Decks	H/S 17	Total	A-up	2-up	3-up	other	EV	SD	N0
8	S	1,081	232	429	254	166	8.1%	1,040	164,761,621
8	H	1,636	510	588	307	231	63.6%	1,279	4,044,541
6	S	1,051	202	459	212	178	5.1%	1,025	404,075,739
6	H	1,573	511	555	309	198	57.3%	1,254	4,790,928

...

Thanks everyone for the replies so far.

Also, I would like to know if there are additional hands to stand on based on the above table and the rules below:
$20 minimum bet on the regular game, and $1 is the only amount you can bet on the side bet,
6 - decks, S17, No Hole Card, double hard 9-11, DAS, split once, and the house edge for the regular game is 0.6612%

Below is one hand to get started and to see if I am correct with my reasoning/answers
note: some of my answers will be off slightly because I am using "infinite deck"

Deviating from basic strategy by standing on a 12 v 2 is the correct play, as deviating gives the player a total gain in EV of $4.089 for that hand and playing correct basic strategy is worth $3.131
Note: When hitting a 12 v 2 there is a 9/13 chance of not busting, when using "infinite deck" reasoning.

Numbers used for working out of 12 v 2 strategy deviation:
Total number of games: 1,000,000,000
Estimated number of times the dealer had any 2-up: 76,923,077
Total number of 8 card 21 with a dealer 2-up: 459
Estimated chance of an 8 card 21 with a dealer 2-up: 1/167,588

Is the above correct or not? If not can you show me the correct answer, and also answer the correct strategy for the hands below:
12 v 3,
12 v 4,
16 v A (I think traditional basic strategy is the correct play for this hand, but I am not sure)
and any other hands you can think of, if any, thanks.

Very quick calculation (based on my list of dealer hands when looking at 6-deck Push22 - Stand soft 17 and no peek) gives pr of 8+ card 21 as .000 001 107 120 355. I'm guessing with Hit soft 17 the chances are bigger since you're turning some 17s into 21s. (My spreadsheet also gives .000 001 163 191 418 for 8 decks.)

Quote: dddkkk1

Thanks everyone for the replies so far.

Also, I would like to know if there are additional hands to stand on based on the above table and the rules below:
<snip>

dddkkk1,

The numbers I gave are based on the assumption that the dealer finishes her hand whether or not any player hands are still active. Your latest post in this thread has me wondering if my assumption is true or not. Can you clarify? For example, if you are playing heads up and you bust or get a BJ, does your sidebet automatically lose?

Dog Hand

^ When analysing sidebets the usual assumption is (since it is a mathematical calculation) that dealer always completes their hand. Some casinos make this clear, other casinos, which would take your sidebet, might not. So it's best to check this with the casino beforehand.

Quote: DogHand

dddkkk1,

The numbers I gave are based on the assumption that the dealer finishes her hand whether or not any player hands are still active. Your latest post in this thread has me wondering if my assumption is true or not. Can you clarify? For example, if you are playing heads up and you bust or get a BJ, does your sidebet automatically lose?

Dog Hand

I asked the casino and they don't play out their hand when the player busts or gets a BJ

Thanks again everyone and sorry for not asking the casino about this before i wrote the OP.

Quote: dddkkk1

I asked the casino and they don't play out their hand when the player busts or gets a BJ

Thanks again everyone and sorry for not asking the casino about this before i wrote the OP.

OK.... ALL the work the members did is useless now..... If this is online, and you are always playing one hand versus the house, then the player advantage certainly goes down a bit. Maybe doghand and the others will redo the math.....

Quote: dddkkk1

Also, I would like to know if there are additional hands to stand on based on the above table and the rules below:
$20 minimum bet on the regular game, and $1 is the only amount you can bet on the side bet,
6 - decks, S17, No Hole Card, double hard 9-11, DAS, split once, and the house edge for the regular game is 0.6612%

Below is one hand to get started and to see if I am correct with my reasoning/answers
note: some of my answers will be off slightly because I am using "infinite deck"

Deviating from basic strategy by standing on a 12 v 2 is the correct play, as deviating gives the player a total gain in EV of $4.089 for that hand and playing correct basic strategy is worth $3.131
Note: When hitting a 12 v 2 there is a 9/13 chance of not busting, when using "infinite deck" reasoning.

Numbers used for working out of 12 v 2 strategy deviation:
Total number of games: 1,000,000,000
Estimated number of times the dealer had any 2-up: 76,923,077
Total number of 8 card 21 with a dealer 2-up: 459
Estimated chance of an 8 card 21 with a dealer 2-up: 1/167,588

Is the above correct or not? If not can you show me the correct answer, and also answer the correct strategy for the hands below:
12 v 3,
12 v 4,
16 v A (I think traditional basic strategy is the correct play for this hand, but I am not sure)
and any other hands you can think of, if any, thanks.

Okay, I understand. The useful concept here is "Equity" of your hand.

When you have a $10 BJ wager and a sidebet of $1 and you are dealt 10-2 vs 2.

Equity of your hand if you STAND:
(1) about $5.97 on you BJ hand, because it has an approximate EV of -0.403
(2) about $5.97 (by coincidence) on your sidebet which has a 1/167,588 chance of winning one million

so your total STAND equity is $5.97 +$5.97 = $11.94

Equity of your hand if you HIT and you are one-on-one with the dealer
(1) about $6.34 on your BJ hand, because it has an approximate EV of -0.366
(2) because hitting a 10-2vs2 has a 31.07% chance of drawing TEN and busting, the equity of your sidebet is $4.11= (1-0.3107)*$5.97

so your Total HIT Equity is about $6.34 + $4.11 = $10.45.

So, in situations in which the dealer will not draw to his upcard 2 when you Bust your 12, you should STAND. If the dealer will draw to his hand anyway,regardless of whether you BUST, then you should HIT.

I haven't yet done the math, but I imagine that you should stand on 12 versus 2-4 (and on 13v2) and on 14,15,16vA if you are betting $5 (or $10) on BJ and $1 on a sidebet, and your chance of making the sidebet will become zero if you BUST your hand (because the dealer will not draw.)

I don't have time right now to do the analysis on all the other hands, maybe someone else will,

Again, sorry about before, on their website they had a lot of the descriptions wrong, eg they had the "player's hand must be unbusted to win the side bet..." for a different game:

But here is the good news, the game is definitely the 6-deck H17 version, again bad description of the side bet made me err on it being S17 just to be safe.
Since it is H17, i think the game is "beatable" now, since the minimum bet is $20 and the house edge is about 0.88% for the main game,