"Unique Blackjack Combinations By Hand Total And Number Of Cards"
April 14th, 2025 at 3:40:20 PM
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So im always prodding at chatgpt. Seeing how it improves, if it ever will.
Today i wanted to know Unique Blackjack Combinations By Hand Total And Number Of Cards based on 6 and 8 deck amounts.
This is my prompt and its repsonses
since my query was essentially garbage to the ai it took some talking to get to a more narrow understanding and it eventually got what i wanted... i think
What im seeing is that, its way easier to get 21 the higher number of cards in 6 and 8 decks
does 98 combinations of 6 card 21s sound right? This seems like a hallucination to me because the 8 and 6 deck amounts are the same in each picture given
chatgpt insists...
Today i wanted to know Unique Blackjack Combinations By Hand Total And Number Of Cards based on 6 and 8 deck amounts.
This is my prompt and its repsonses
Quote: heatmapin a six - eight deck blackjack game list all of the combinations of cards that will make 21 up to a 10 card hand
since my query was essentially garbage to the ai it took some talking to get to a more narrow understanding and it eventually got what i wanted... i think
Quote:Here's the updated chart reflecting realistic card limits from 6 decks (e.g., only 24 Aces and 96 Tens available). The counts are now based on valid combinations that could physically be dealt within that constraint. Let me know if you want the same for 8 decks or a specific total exported.
6 deck
8 deck
What im seeing is that, its way easier to get 21 the higher number of cards in 6 and 8 decks
does 98 combinations of 6 card 21s sound right? This seems like a hallucination to me because the 8 and 6 deck amounts are the same in each picture given
chatgpt insists...
Quote: heatmapsomething seems off about your 6 and 8 deck charts in the sense that 98 combinations of 6 card hands that equal 21 does not seem possible for both
Quote:ChatGPT said:
You're right to question it — it does seem surprising that both the 6-deck and 8-deck environments yield exactly 98 unique 6-card combinations that total 21.
But here's why it's actually possible:
The limit on card counts (e.g., 24 of each value in 6 decks vs 32 in 8 decks) only begins to matter when a combination requires more instances of a value than exist in the shoe.
A 6-card hand has relatively low demand on card count variety. In nearly all valid combinations of 6 cards totaling 21, no single card value is used more than 4 or 5 times — well below both decks' limits.
So in this case:
All 98 valid 6-card combinations that sum to 21 can be formed using only what's available in a 6-deck shoe.
Increasing to 8 decks doesn’t introduce new combinations — it just provides more ways to realize them, which we are not counting (we count only unique value sets, not permutations or frequency of appearance).
If you'd like, I can list the 98 combinations for inspection — or highlight which hand sizes start to diverge between deck sizes.
Last edited by: heatmap on Apr 14, 2025
April 15th, 2025 at 7:45:46 AM
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Your question is too complicated. Simplify it.
What is the probability of a 3-card 21 hand?
What is the probability of a 4-card 21 hand?
What is the probability of a 5 or more card 21 hand?
What is the probability of a 3-card 21 hand?
What is the probability of a 4-card 21 hand?
What is the probability of a 5 or more card 21 hand?
April 15th, 2025 at 9:16:56 AM
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are your combinations sequence-specific or sequence-independent? (other than the proviso that hands making a soft 17-20 are no further hit.)
Are
2-3-7-A-8
and
3-2-7-A-8
counted as two combinations? or the same combination?
Are
2-3-7-A-8
and
3-2-7-A-8
counted as two combinations? or the same combination?
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
April 16th, 2025 at 6:59:51 PM
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Assuming you count 237AA7 the same as 327AA7 then I can see 86 ways to get to 21 with 6 cards. However it's late here so I haven't checked it. (I also assume the dealer hits soft 17.) "0" means a Ten card.
1: 111116 T:21
2: 211115 T:21
3: 311114 T:21
4: 110117 T:21
5: 111918 T:21
6: 221114 T:21
7: 021116 T:21
8: 921117 T:21
9: 211818 T:21
10: 031115 T:21
11: 931116 T:21
12: 831117 T:21
13: 941115 T:21
14: 841116 T:21
15: 741117 T:21
16: 851115 T:21
17: 761115 T:21
18: 661116 T:21
19: 022115 T:21
20: 922116 T:21
21: 822117 T:21
22: 932115 T:21
23: 832116 T:21
24: 732117 T:21
25: 842115 T:21
26: 742116 T:21
27: 752115 T:21
28: 662115 T:21
29: 833115 T:21
30: 733116 T:21
31: 443118 T:21
32: 743115 T:21
33: 643116 T:21
34: 653115 T:21
35: 444117 T:21
36: 644115 T:21
37: 554115 T:21
38: 222221 T:21
39: 922215 T:21
40: 822216 T:21
41: 722217 T:21
42: 432219 T:21
43: 832215 T:21
44: 732216 T:21
45: 442218 T:21
46: 742215 T:21
47: 642216 T:21
48: 652215 T:21
49: 333219 T:21
50: 433218 T:21
51: 733215 T:21
52: 633216 T:21
53: 443217 T:21
54: 643215 T:21
55: 553215 T:21
56: 444216 T:21
57: 544215 T:21
58: 333318 T:21
59: 433317 T:21
60: 633315 T:21
61: 443316 T:21
62: 543315 T:21
63: 444315 T:21
64: 322220 T:21
65: 422229 T:21
66: 822225 T:21
67: 722226 T:21
68: 332229 T:21
69: 432228 T:21
70: 732225 T:21
71: 632226 T:21
72: 442227 T:21
73: 642225 T:21
74: 552225 T:21
75: 333228 T:21
76: 433227 T:21
77: 633225 T:21
78: 443226 T:21
79: 543225 T:21
80: 444225 T:21
81: 333327 T:21
82: 433326 T:21
83: 533325 T:21
84: 443325 T:21
85: 333336 T:21
86: 433335 T:21
April 17th, 2025 at 9:34:38 AM
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Wonderful! Of these 86, there are three combinations, each using more than four cards of the same rank, meaning they cannot happen in a single deck game:
111116,
222221,
333336.
Also, this consideration is more related to the bonus payout in a Spanish 21 game, in which all 10s are removed from the deck. If it's possible, could you please calculate their probabilities in an eight-deck Spanish 21 game?
111116,
222221,
333336.
Also, this consideration is more related to the bonus payout in a Spanish 21 game, in which all 10s are removed from the deck. If it's possible, could you please calculate their probabilities in an eight-deck Spanish 21 game?
Last edited by: aceside on Apr 17, 2025
April 17th, 2025 at 10:22:11 AM
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^ I have no idea what the probabilities are. There are some games where the bonus for getting 21 is so great you would even hit some 20s. In a standard BJ game a hand such as AAAAA6 could only happen with EHNC where you didn't split Aces and then the 6 has to come last. Similarly with 222221, the Ace has to come last. Also 333444 is impossible if you don't hit hard 17.
If you really want the probabilities then you have to calculate the correct strategy for all the possible player hands (which don't exceed 21) and then work through all the possible series of cards where the player takes more cards (taking into account splits and doubles). Note that if there are bonuses for 21s then these have to be factored in during the strategy analysis. (Note if this is a side-bet then the ratio of base bet to side-bet will affect this. In practice you either have to set a limit or set the bonus so small the game can't be beaten.)
Also note that when working out House Edge one is usually only interested in EVs of the possible player hands and deciding whether standing or hitting (or Dbl/Spl/Surr) by comparing the stand EV to the range of other EVs (if progressing by hitting or doubling). This process is recursive, thus does not evaluate the actual chance of getting something like 6-card 21s.
If you really want the probabilities then you have to calculate the correct strategy for all the possible player hands (which don't exceed 21) and then work through all the possible series of cards where the player takes more cards (taking into account splits and doubles). Note that if there are bonuses for 21s then these have to be factored in during the strategy analysis. (Note if this is a side-bet then the ratio of base bet to side-bet will affect this. In practice you either have to set a limit or set the bonus so small the game can't be beaten.)
Also note that when working out House Edge one is usually only interested in EVs of the possible player hands and deciding whether standing or hitting (or Dbl/Spl/Surr) by comparing the stand EV to the range of other EVs (if progressing by hitting or doubling). This process is recursive, thus does not evaluate the actual chance of getting something like 6-card 21s.
April 17th, 2025 at 10:40:25 AM
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But in a Spanish 21 game, these probability numbers for these bonus 5-card, 6-card, 7-card 21 hands are all listed out. I guess they come out of simulations.
April 17th, 2025 at 10:54:25 AM
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^ Given it's part of the base game then the effect of the bonuses would easily be factored into the EVs. They have to do the EV analysis, so It's not that hard to also work out the probabilities. It's slightly complicated by the addition of suits, if I recall Spades and non-Spades, but this might only affect the decision for 2-card hands. Also if splitting/doubling affects bonuses then this would need to be included.
It's not impossible to do and, relative to some poker games, quite easy!
I know the Wizard quite often lists the various outcomes for BJ (e.g. +8 thru -8) with their combinations and contributions to EV, so it's all possible to calculate.
It's not impossible to do and, relative to some poker games, quite easy!
I know the Wizard quite often lists the various outcomes for BJ (e.g. +8 thru -8) with their combinations and contributions to EV, so it's all possible to calculate.
April 18th, 2025 at 7:41:52 AM
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I still haven’t figured out how to handle this. Let’s just simplify this problem to these three hands only:
AAAAA6,
22222A,
333336.
What is the probability of each of these hands for a dealer in a six-deck Hit-17 Spanish 21 game?
AAAAA6,
22222A,
333336.
What is the probability of each of these hands for a dealer in a six-deck Hit-17 Spanish 21 game?
April 18th, 2025 at 8:01:42 AM
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In these cases the cards have to come out in that specified order so it's
(a) 24/288 (24 Aces, 288 cards left in the shoe)
(b) 23/287 (one less Ace now available, one less in the shoe)
...
(f) 24/283 (24 Sixes, 283 = 288, less 5 cards already gone, now left in the shoe).
Multiply (a)(b)...(f) or 24*23*22*21*20*24 / (288*287*286*285*284*283)
If you had a hand such as 443226 then you have to work out the various ways the first five cards could come out, and use 24*23 24 24*23 P / (288...283).
Note : If this is a side-bet based on the Dealer's result then you may have to consider the occasions when the Dealer does not/usually would not draw additional cards. This could be because the Player had a BJ or because the Player busted. Alternatively insist they need to resolve any side-bets even if there are no Player's hands left.
(a) 24/288 (24 Aces, 288 cards left in the shoe)
(b) 23/287 (one less Ace now available, one less in the shoe)
...
(f) 24/283 (24 Sixes, 283 = 288, less 5 cards already gone, now left in the shoe).
Multiply (a)(b)...(f) or 24*23*22*21*20*24 / (288*287*286*285*284*283)
If you had a hand such as 443226 then you have to work out the various ways the first five cards could come out, and use 24*23 24 24*23 P / (288...283).
Note : If this is a side-bet based on the Dealer's result then you may have to consider the occasions when the Dealer does not/usually would not draw additional cards. This could be because the Player had a BJ or because the Player busted. Alternatively insist they need to resolve any side-bets even if there are no Player's hands left.
April 18th, 2025 at 8:07:50 AM
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This doesn’t sound correct, because a TTTTTT hand is impossible. A hand that is composed of three Tens is also impossible.
April 18th, 2025 at 8:13:25 AM
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Where the dealer doesn't take a card you stop the calculation, so you would only have 72/288 * 71/287. You could consider this as 72 71 286 285 284 283 / (288 287 286 285 284 283) if it's easier, essentially it doesn't matter what the third card down is, so all 286 count..
April 18th, 2025 at 9:34:08 AM
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in the past Miplet has shared some excel files with all the combinations and probabilities of multi-card blackjack hands Don't remember if the doubling and split pair strategies were worked into that.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
April 18th, 2025 at 4:27:48 PM
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Quote: charliepatrickWhere the dealer doesn't take a card you stop the calculation, so you would only have 72/288 * 71/287. You could consider this as 72 71 286 285 284 283 / (288 287 286 285 284 283) if it's easier, essentially it doesn't matter what the third card down is, so all 286 count..
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I’ve read your post a few times but still haven’t understood what you mean by 72/288 * 71/287. Is it possible you simulate the result out first and then we calculate it using combination and permutation to match the result?
April 19th, 2025 at 1:49:15 AM
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You had originally asked for these based on six decks (and then seem to have edited it to eight). The 288 comes from 6 decks times 48 cards in each deck. The 72 comes from 6 decks each with four K,Q,J. Thus when the first card is dealt 72 out of 288 are Ten-valued cards. When the second card is dealt it's 71 out of 287.
April 19th, 2025 at 10:27:37 AM
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Quote: acesideQuote: charliepatrickWhere the dealer doesn't take a card you stop the calculation, so you would only have 72/288 * 71/287. You could consider this as 72 71 286 285 284 283 / (288 287 286 285 284 283) if it's easier, essentially it doesn't matter what the third card down is, so all 286 count..
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I’ve read your post a few times but still haven’t understood what you mean by 72/288 * 71/287. Is it possible you simulate the result out first and then we calculate it using combination and permutation to match the result?
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You keep mentioning simulations. Simulations that accurately predict a low probability out to six significant digits are very hard to do- the number of trials would be quite large. The kinds of calculations for probability of individual hands are always done by using elementary combination theory. It's very simple - almost trivial - to do. Simulations tend to be for calculating house edge - by simulating gameplay.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
April 19th, 2025 at 10:57:09 AM
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https://miplet.net/blackjack/ and download the reflex excel file. I list all dealer hands. You can edit the deck composition to all black cards. Column C on the h17/s17 sheets will now show dealer probabilities assuming they always play out their hand.
“Man Babes” #AxelFabulous
April 19th, 2025 at 10:57:25 AM
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Quote: miplethttps://miplet.net/blackjack/ and download the reflex excel file. I list all dealer hands. You can edit the deck composition to all black cards. Column C on the h17/s17 sheets will now show dealer probabilities assuming they always play out their hand.
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I've looked into this file. It lists these hands and their probability numbers:
10, 7: 0.011880012
10, 8: 0.011880012
10, 9: 0.011880012
10, 10: 0.046926047
Why are these numbers about half of what they should be? It looks like you intentionally made them half, because I see this caption (Hand/Prob 0.5). For example, this two-card hand of (10, 10) should have a probability about
(4/13)x(4/13)= 0.0947.
Anyway, I've figured out this part with an infinite deck game. Thank you all!
Last edited by: aceside on Apr 19, 2025
April 20th, 2025 at 11:50:59 AM
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It’s because it’s set up for reflex, a side bet on the number of consecutive red cards in the dealer’s hand. If you change the deck composition to only black cards then the dealer will only get black cards so all probabilities will be in column C.
“Man Babes” #AxelFabulous
April 20th, 2025 at 4:04:36 PM
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So does this info have any practical application?
April 20th, 2025 at 6:33:18 PM
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Using miplet spreadsheets, I find these probabilities for a 6-card 21 hand:
0.038%, for six-deck Blackjack, that is, 1/2629;
0.059%, for six-deck Spanish 21, that is, 1/1670.
These numbers assume you play like a Hit-17 dealer does all the time. In reality, these numbers should be smaller because player hits less. I think there is not much use, but it seems miplet has researched this.
0.038%, for six-deck Blackjack, that is, 1/2629;
0.059%, for six-deck Spanish 21, that is, 1/1670.
These numbers assume you play like a Hit-17 dealer does all the time. In reality, these numbers should be smaller because player hits less. I think there is not much use, but it seems miplet has researched this.
April 21st, 2025 at 6:37:33 PM
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i specifically brought this up because i was thinking of craps probability amounts and wondered ... this complex thing about blackjack and if chatgpt could handle it... its got the jist of what i wanted to see but seemed a bit off with the amounts but they are seemingly close based on some of the discussions...
i also in my mind think i see alot of 6+ card hands digitally and physically and it seemed odd but when you look at the combos it doesnt (if they are correct)
i have to say i think chatgpt kind of knows "me" and knows i want to see a heatmap of things but it did not explicitly tell me it was going to make it and just assumed or made it itself
i also in my mind think i see alot of 6+ card hands digitally and physically and it seemed odd but when you look at the combos it doesnt (if they are correct)
i have to say i think chatgpt kind of knows "me" and knows i want to see a heatmap of things but it did not explicitly tell me it was going to make it and just assumed or made it itself
April 21st, 2025 at 11:36:38 PM
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Spanish 21 pays a bonus for 5 card and 6 card 21's and they don't hit often. It's just a mirage. Sure there's more ways to hit 21 with 6 cards than 2 cards, but you're more likely to get a BJ than a 6 card 21 because of the extra 10's in the deck (compared to there being no face cards and only 10's in the deck). Of course Dealers are free to hit 21 with as many cards as they wish just to bust the whole table minus the 21 winners.
April 22nd, 2025 at 8:40:38 AM
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I’ve tried ChatGPT using OP questions. It looks like it’s smarter than me. Not only did it quickly give me a possible correct answer of 98 combinations, but also a Python program for this task. We human answered OP’s original question by assuming the constraint of player playing as a dealer, but this is not in the original query.
Last edited by: aceside on Apr 22, 2025
April 22nd, 2025 at 11:45:38 AM
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I also get 98 combinations changing the code to stop only when it reaches 21 (or more) - it actually doesn't matter whether you allow the player to hit soft 21 or not as there will always be a similar combination of cards, say the Aces come out last, where you hit hard totals before the string of Aces. Of course this affects the chances of the player making 21 with some combinations as, for instance, they probably wouldn't be allowed to hit BJ.
