tooncestdc
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June 24th, 2015 at 3:50:30 AM permalink
I've seen the odds of a dealer BJ as about .000024. And I've seen returns of 7-card 21 bonuses that paid 3-1, 5-1, etc. But let's say there was a promotion that paid 50-1 or 100-1 on a 7-card 21. How high could the chances of drawing a 7-card 21 get as you care less and less about winning your basic bet? What are common strategy changes in addition to obvious ones like hitting all 6-card 20 or less.

Curious for both regular shoe BJ as well as Spanish.
ThatDonGuy

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June 24th, 2015 at 6:54:17 AM permalink
If you play 6-deck regular 21 and always hit if there is a chance of a 7-card (or more) 21 (e.g. hit on 20 with 6 or more cards, but not with 5 or fewer), the probability of a 7-card or higher 21 is about 1 / 1800.

The major change needed from basic strategy (after "Hit 20 with 6 or more cards") appears to be, "Never split Aces." A soft 12 is more likely to lead to a seven-card 21 than what you might end up with from your split.
tooncestdc
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June 24th, 2015 at 11:01:16 AM permalink
When I've been trying, some of the strategy changes I've done is hit 4 card soft 19 vs. 9 through A, hit some 4-card soft 20s, hit all 5-card soft totals and some 5-card 17s or even smaller vs a 2-6. I hadn't tried not splitting aces. You are likely right, but I know hitting aces gives up a lot of EV and I wonder if 2-card totals are too low to make changes hoping for a 7-card 21. I've noticed that it gives up a lot more EV in Spanish 21 than regular BJ, since you lose the opportunity to double 3 and 4 card hands that total 9-11, or doubling and redoubling soft totals.

For example, there is a casino dealing spanish 21, where the first 7 card 21 is worth \$500 with a \$3 minimum bet plus 21 cents of -EV per required side betting. Given what you said above, I'm ballparking that the true odds (in Spanish 21) with all rational deviations would be about 1 in 500, but I'm not sure what it would cost me in EV on the base bet.
Viper21
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May 14th, 2023 at 12:57:48 PM permalink
I am looking for the probability of a 7 card 21 with strategy changes on an 8 deck CSM (not to be confused with 7 card Charlie). I have an 8 deck game that is paying 500 for 1 for a 7 card 21.

If I am understanding DonGuy's post, for a 6 deck, strategy change of hitting any 6 card 20 and never splitting Aces it would have a probability of 0.00055556. This is way less than a dealer's 7 card 21 being 0.000029251 or 1 / 34,186 which makes sense that it would be less but that much less?

So is the probability of a 6 deck 7 card 21 0.00055556 or if not can someone help me figure out the probability of an 8 deck 7 card 21?

Thank you
DRich
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May 14th, 2023 at 4:18:24 PM permalink
I would think that even hitting a 5 card 18 and probably a 5 card 19 against a 10 would be profitable.
At my age, a "Life In Prison" sentence is not much of a deterrent.
DogHand
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May 14th, 2023 at 10:21:32 PM permalink
Quote: tooncestdc

I've seen the odds of a dealer BJ as about .000024.<snip>link to original post

tooncestdc,

WHAT???? Where did you see the above-quoted figure?

The dealer averages a BJ about once in 21 rounds: off the top of a 6D shoe, the dealer's BJ probability is 2*96*24/(312*311) = 0.047489...

Dog Hand
Viper21
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May 14th, 2023 at 10:35:58 PM permalink
It's an old post so not sure they will reply but pretty sure they meant a dealer 7 card 21 as wizard has a chart stating double deck dealer 7 card 21 being 0.0000234897.

Thanks
charliepatrick
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May 15th, 2023 at 3:03:39 PM permalink
If it paid 500/1 then it is probably worth considering splitting some pairs where you might not have done so. Note in my quick look, I did not include the effects of being ablt to double and just assumed you could hit or stand.

if the bonus was paid on either hand and say you couldn't/wouldn't double after split, then my estimate was you'd split 10(789) 9832(4 6789) 6 (4 678) 654(456789) A(6789). Also I can imagine some close doubling might be a bad idea since the cost of not being able to hit 3-card 12+ is quite high. Also not splitting 3s vs low cards is a fairly close decision, so splitting 11 might make it worth it.

Similarly a quick look vs 10 you stand 2-card 17+ 2-card soft 20, whereas vs 6 you do stand 2-card 13+ 3-card 15+ 4-card 17+ and 2-card soft 20. I think you probably hit any 3-card soft total.
Viper21
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May 15th, 2023 at 5:15:22 PM permalink
I appreciate the look at strategy changes. What about if you just went for it every single hand ignoring the main bet win/loss? So basically hit until you either bust or reach a hard 21 like what DonGuy posted. What would be the probability of reaching a 7 card 21.
AxelWolf
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May 15th, 2023 at 6:17:50 PM permalink
Quote: Viper21

I appreciate the look at strategy changes. What about if you just went for it every single hand ignoring the main bet win/loss? So basically hit until you either bust or reach a hard 21 like what DonGuy posted. What would be the probability of reaching a 7 card 21.
link to original post

FFS STOP JACKIINNG AROUND...what's the promo and where? (-:
♪♪Now you swear and kick and beg us That you're not a gamblin' man Then you find you're back in Vegas With a handle in your hand♪♪ Your black cards can make you money So you hide them when you're able In the land of casinos and money You must put them on the table♪♪ You go back Jack do it again roulette wheels turinin' 'round and 'round♪♪ You go back Jack do it again♪♪
gordonm888
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May 16th, 2023 at 11:05:33 AM permalink
Quote: Viper21

I appreciate the look at strategy changes. What about if you just went for it every single hand ignoring the main bet win/loss? So basically hit until you either bust or reach a hard 21 like what DonGuy posted. What would be the probability of reaching a 7 card 21.
link to original post

To calculate this probability, we could ignore the dealer's card and simply look at all the ways that seven cards with possible values of 1-11 can be combined to add up to 21, and then calculate the probability of each combination and the number of distinct permutations of each combination (to represent the different sequences in which each set of seven cards can be dealt.)

I don't think this would be very hard.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
Wizard
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May 16th, 2023 at 11:41:38 AM permalink
Quote: tooncestdc

I've seen the odds of a dealer BJ as about .000024. And I've seen returns of 7-card 21 bonuses that paid 3-1, 5-1, etc. But let's say there was a promotion that paid 50-1 or 100-1 on a 7-card 21. How high could the chances of drawing a 7-card 21 get as you care less and less about winning your basic bet? What are common strategy changes in addition to obvious ones like hitting all 6-card 20 or less.

Curious for both regular shoe BJ as well as Spanish.
link to original post

Give me the exact rules of the promotion and who is running it and I'd be happy to provide a full strategy.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
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May 16th, 2023 at 12:09:18 PM permalink
I'm in a good mood today and will go a little ways with this, not knowing the details. Based on a quick simulation, there are probabilities of reaching 5 to 7 cards, with a goal of getting to 21 or busting:

5 cards = 1 in 67
6 cards = 1 in 280
7 cards = 1 in 1594

This was just a 10-second simulation. I'll let it run a few hours to see how it changes things.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
DRich
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May 16th, 2023 at 12:20:09 PM permalink
Quote: Wizard

I'm in a good mood today and will go a little ways with this, not knowing the details. Based on a quick simulation, there are probabilities of reaching 5 to 7 cards, with a goal of getting to 21 or busting:

5 cards = 1 in 67
6 cards = 1 in 280
7 cards = 1 in 1594

This was just a 10-second simulation. I'll let it run a few hours to see how it changes things.
link to original post

Make sure your simulations are treating an Ace as one and not eleven so you can continue to draw more cards. If you have a six card 21 using an ace as eleven obviously you can dar another card with a good chance of getting 500-1.
At my age, a "Life In Prison" sentence is not much of a deterrent.
charliepatrick
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May 16th, 2023 at 1:53:17 PM permalink
Using infinite decks and looping code (e.g for card1=10 to 1; for card2=10 to 1... if total=21 yippee, if total>21 bust, else move onto next card) but giving in if one gets to 10 cards. As has been said if on 7 cards or more the total is 11 and there's an Ace somewhere count it as a 21.
 No of cards BUST 21 3 .382 339 553 937 .078 288 575 330 1 in 12.773 4 .306 711 949 862 .042 015 335 597 1 in 23.801 5 .128 728 524 373 .015 068 961 709 1 in 66.362 6 .034 456 304 362 .003 681 935 622 1 in 271.596 7 .006 737 944 102 .000 681 960 340 1 in 1 466.361 8 .001 034 753 232 .000 099 150 367 1 in 10 085.691 9 .000 127 998 499 .000 011 927 013 1 in 83 843.289 10 .000 015 125 655 < Hands not resolved

Guessing that 1 hand in 12 with 10+cards makes 21, this makes the total about .000 794 298 191 or 1 hand in 1259 being a 7-card+ 21.
Wizard
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May 16th, 2023 at 5:11:29 PM permalink
Quote: DRich

Make sure your simulations are treating an Ace as one and not eleven so you can continue to draw more cards. If you have a six card 21 using an ace as eleven obviously you can dar another card with a good chance of getting 500-1.
link to original post

I am. I even have the player hitting a blackjack, because he only has 11. The simulation absolutely always hits to 21, even with a 10-10, where there is no hope of a 5+ card 21.

Here are the results of a longer simulation. The "cards" shows the number of cards in the 21-point hand.

Cards Count Probability Inverse
Bust 52,104,124,978 0.859949 1.16
3 4,759,037,984 0.078545 12.73
4 2,557,594,660 0.042212 23.69
5 908,819,311 0.015000 66.67
6 216,326,234 0.003570 280.09
7 38,049,196 0.000628 1,592
8 5,220,188 0.000086 11,607
9 572,119 0.000009 105,904
10 50,292 0.000001 1,204,760
11 3,487 0.000000 17,375,910
12 192 0.000000 315,571,868
13 14 0.000000 4,327,842,761
Total 60,589,798,655 1.000000

p.s. My simulation is based on six decks.
Last edited by: Wizard on May 17, 2023
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThatDonGuy

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May 16th, 2023 at 5:45:36 PM permalink
Using an infinite deck, I get an exact probability (which matches simulation) of getting 21 with 7 or more cards as:

195,707,534,756,955,275,874 / 247,064,529,073,450,392,704,413

or 1 / 1262.417.

This assumes (a) you hit on all hands, (b) you do not double or split, and (c) if you are dealt a blackjack, you are allowed to treat it as an 11 and hit.
DRich
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May 16th, 2023 at 6:01:31 PM permalink
Quote: ThatDonGuy

Using an infinite deck, I get an exact probability (which matches simulation) of getting 21 with 7 or more cards as:

195,707,534,756,955,275,874 / 247,064,529,073,450,392,704,413

or 1 / 1262.417.

This assumes (a) you hit on all hands, (b) you do not double or split, and (c) if you are dealt a blackjack, you are allowed to treat it as an 11 and hit.
link to original post

I would assume that splitting bigger cards would be correct.
At my age, a "Life In Prison" sentence is not much of a deterrent.
DogHand
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May 16th, 2023 at 10:38:53 PM permalink
Quote: DRich

Quote: ThatDonGuy

Using an infinite deck, I get an exact probability (which matches simulation) of getting 21 with 7 or more cards as:

195,707,534,756,955,275,874 / 247,064,529,073,450,392,704,413

or 1 / 1262.417.

This assumes (a) you hit on all hands, (b) you do not double or split, and (c) if you are dealt a blackjack, you are allowed to treat it as an 11 and hit.
link to original post

I would assume that splitting bigger cards would be correct.
link to original post

DRich,

Good point! To have ANY chance of a 7-card 21, you would have to split 9's and all 10's.

8-8 has the chance of drawing 5 consecutive Aces, but I would guess that splitting gives a better chance of reaching the goal.

7-7 is probably also a split: do you have a better chance of drawing 5 cards totaling 7, or two chances at 6 cards totaling 14?

On the low end, I would guess that splitting Aces, deuces, 3's, and 4's would be incorrect.

5's and 6's are tough to guess.

Dog Hand
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