Math question, How many cards does the dealer draw with up card 2
May 28th, 2025 at 6:25:47 PM
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Hi, I've got a math question
How many cards does the dealer draw on average if they have a up card 2? (Not including the hole card they already have)
Is there a significant difference between S17 and H17?
Is there a significant difference between 2 Decks and 6 Decks?
Thanks!
(Would prefer calculated answers over simulated answers, but all answers are welcomed!)
How many cards does the dealer draw on average if they have a up card 2? (Not including the hole card they already have)
Is there a significant difference between S17 and H17?
Is there a significant difference between 2 Decks and 6 Decks?
Thanks!
(Would prefer calculated answers over simulated answers, but all answers are welcomed!)
May 28th, 2025 at 7:52:28 PM
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I don't know the answer but let me point to the right direction. In this thread, "Unique Blackjack Combinations By Hand Total And Number Of Cards," miplet shares an Excel spreadsheet to figure this out.
May 28th, 2025 at 8:20:20 PM
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Quote: KatrinaOHi, I've got a math question
How many cards does the dealer draw on average if they have a up card 2? (Not including the hole card they already have)
Is there a significant difference between S17 and H17?
Is there a significant difference between 2 Decks and 6 Decks?
Thanks!
(Would prefer calculated answers over simulated answers, but all answers are welcomed!)
link to original post
The numbers do not include the first 2 cards in the dealer's hand - and those fractions are exact
Hit Soft 17:
1 deck: 1.6828 (25,002,425,431 / 14,857,804,500)
2 decks: 1.7045 (4,620,039,689,933 / 2,710,487,044,188)
3 decks: 1.7118 (589,494,362,017,745,141 / 344,373,782,121,172,650)
4 decks: 1.7154 (6,968,823,981,140,719 / 4,062,427,646,249,763)
5 decks: 1.7176 (13,588,350,138,862,680,919 / 7,911,134,273,875,250,880)
6 decks: 1.7191 (1,818,894,610,413,553,006,409 / 1,058,060,120,624,333,361,150)
Stand Soft 17:
1 deck: 1.6618 (74,071,308,337 / 44,573,413,500)
2 decks: 1.6834 (22,814,081,349,041 / 13,552,435,220,940)
3 decks: 1.6906 (531,693,731,743,127 / 314,496,604,676,870)
4 decks: 1.6942 (103,240,549,938,714,961 / 60,936,414,693,746,445)
5 decks: 1.6964 (244,008,792,631,546,973 / 143,838,804,979,550,016)
6 decks: 1.6979 (496,800,545,418,173 / 292,605,364,612,950)
May 28th, 2025 at 8:28:00 PM
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This is lightning speed! Mind you do a dealer upcard Ace?
May 29th, 2025 at 7:53:32 AM
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Quote: acesideThis is lightning speed! Mind you do a dealer upcard Ace?
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Upcard Ace - again, this is how many cards after the first two
Note that two-card 21s are not included in the calculations
Hit Soft 17:
1 deck: 1.2006 (71,035,117 / 59,167,500)
2 decks: 1.2225 (4,078,784,930,567 / 3,336,417,824,850)
3 decks: 1.2299 (173,036,487,383,007,301 / 140,696,753,906,780,610)
4 decks: 1.2335 (2,435,631,444,535,164,822 / 1,974,510,856,150,829,425)
5 decks: 1.2357 (2,803,949,618,270,529,524,069 / 2,269,029,380,443,332,091,200)
6 decks: 1.2372 (6,485,656,997,795,512,854,831 / 5,242,103,866,651,640,902,375)
Stand Soft 17:
1 deck: 0.9730 (19,189,057 / 19,722,500)
2 decks: 0.9921 (308,939,249,891 / 311,398,996,986)
3 decks: 0.9985 (20,971,890,984,179 / 21,002,650,232,390)
4 decks: 1.0018 (42,170,354,069,631,577 / 42,096,170,537,225,805)
5 decks: 1.0037 (831,481,505,521,305,557 / 828,415,253,904,100,800)
6 decks: 1.0050 (6,448,335,365,080,558,674 / 6,416,283,802,511,188,375)
Upcard 3
Hit Soft 17:
1 deck: 1.5685 (1,942,055,607/1,238,150,375)
2 decks: 1.5859 (42,142,822,669/26,573,402,394)
3 decks: 1.5916 (1,251,410,746,247,854/786,241,511,692,175)
4 decks: 1.5945 (485,813,228,474,984,792/304,682,073,468,732,225)
5 decks: 1.5962 (6,377,661,552,127,251/3,995,522,360,543,056)
6 decks: 1.5973 (1,865,433,119,204,080,761/1,167,836,777,731,052,275)
Stand Soft 17:
1 deck: 1.5530 (69,223,442,807/44,573,413,500)
2 decks: 1.5685 (53,142,479,907,967/33,881,088,052,350)
3 decks: 1.5736 (2,474,505,648,580,349/1,572,483,023,384,350)
4 decks: 1.5762 (96,047,290,219,604,908/60,936,414,693,746,445)
5 decks: 1.5777 (2,496,314,902,694,317,721/1,582,226,854,775,050,176)
6 decks: 1.5787 (1,843,715,612,708,618,851/1,167,836,777,731,052,275)
__________________________________________________
Upcard 4
Hit Soft 17:
1 deck: 1.4684 (2,337,560,018/1,591,907,625)
2 decks: 1.4759 (375,971,533,229/254,745,022,950)
3 decks: 1.4784 (50,832,113,868,347/34,383,739,578,375)
4 decks: 1.4796 (151,027,383,588,383/102,071,046,388,185)
5 decks: 1.4804 (2,332,969,508,026,097/1,575,923,162,126,544)
6 decks: 1.4809 (7,338,466,136,631,638/4,955,460,160,103,475)
Stand Soft 17:
1 deck: 1.4497 (1,794,980,554/1,238,150,375)
2 decks: 1.4577 (866,441,252,773/594,405,053,550)
3 decks: 1.4603 (117,158,509,373,623/80,228,725,682,875)
4 decks: 1.4616 (12,787,713,077,135/8,748,946,833,273)
5 decks: 1.4624 (256,073,936,044,533/175,102,573,569,616)
6 decks: 1.4630 (50,747,187,008,014,886/34,688,221,120,724,325)
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Upcard 5
Hit Soft 17:
1 deck: 1.3585 (262,128,569/192,958,500)
2 decks: 1.3628 (405,013,798,726/297,202,526,775)
3 decks: 1.3642 (11,752,711,461/8,615,165,174)
4 decks: 1.3649 (417,952,641,435,986/306,213,139,164,555)
5 decks: 1.3653 (2,868,893,063,721,737/2,101,230,882,835,392)
6 decks: 1.3656 (47,371,178,870,967,176/34,688,221,120,724,325)
Stand Soft 17:
1 deck: 1.3517 (1,825,758,313/1,350,709,500)
2 decks: 1.3551 (1,208,181,305,434/891,607,580,325)
3 decks: 1.3562 (365,734,320,581/269,676,388,850)
4 decks: 1.3568 (59,351,502,358,238/43,744,734,166,365)
5 decks: 1.3571 (8,554,825,111,431,461/6,303,692,648,506,176)
6 decks: 1.3573 (6,486,268,188,784/4,778,650,106,175)
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Upcard 6
Hit Soft 17:
1 deck: 1.2544 (5,083,080,739/4,052,128,500)
2 decks: 1.2584 (1,122,007,736,027/891,607,580,325)
3 decks: 1.2597 (3,277,694,743,151/2,602,012,724,850)
4 decks: 1.2603 (4,435,862,569,153/3,519,691,254,765)
5 decks: 1.2607 (1,135,266,632,078,911/900,527,521,215,168)
6 decks: 1.2609 (2,302,044,385,925,429/1,825,695,848,459,175)
Stand Soft 17:
1 deck: 1.1417 (4,626,204,139/4,052,128,500)
2 decks: 1.1463 (146,010,001,253/127,372,511,475)
3 decks: 1.1478 (119,464,701,911/104,080,508,994)
4 decks: 1.1485 (13,025,888,174,597/11,341,227,376,465)
5 decks: 1.1490 (1,034,687,238,129,943/900,527,521,215,168)
6 decks: 1.1493 (123,424,361,687,437/107,393,873,438,775)
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Note that for upcards 7 through 10, a Soft 17 is not possible
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Upcard 7
1 deck: 0.7826 (211,415,219/270,141,900)
2 decks: 0.7871 (7,088,363,689/9,006,137,175)
3 decks: 0.7885 (136,780,756,207/173,467,514,990)
4 decks: 0.7892 (80,557,349,596,333/102,071,046,388,185)
5 decks: 0.7897 (237,036,127,683,931/300,175,840,405,056)
6 decks: 0.7899 (288,439,135,922,401/365,139,169,691,835)
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Upcard 8
1 deck: 0.6639 (179,337,791/270,141,900)
2 decks: 0.6677 (4,050,857/6,067,215)
3 decks: 0.6689 (11,681,250,697/17,463,172,650)
4 decks: 0.6695 (145,712,606,377/217,635,493,365)
5 decks: 0.6699 (794,808,220,067/1,186,465,772,352)
6 decks: 0.6701 (191,018,681,417/285,042,287,035)
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Upcard 9
1 deck: 0.5704 (3,350,047/5,872,650)
2 decks: 0.5651 (577,192,744/1,021,314,525)
3 decks: 0.5634 (258,914,993/459,557,175)
4 decks: 0.5625 (122,426,943,872/217,635,493,365)
5 decks: 0.5620 (47,629,137,395/84,747,555,168)
6 decks: 0.5617 (1,120,689,329,448/1,995,296,009,245)
__________________________________________________
Upcard 10
(2-card 21s not included)
1 deck: 0.4948 (382,587/773,150)
2 decks: 0.4903 (197,949,344/403,709,625)
3 decks: 0.4888 (787,578,049/1,611,118,509)
4 decks: 0.4881 (970,464,336/1,988,251,745)
5 decks: 0.4877 (123,647,305/253,554,144)
6 decks: 0.4874 (52,787,514,928/108,312,834,245)
Last edited by: ThatDonGuy on May 29, 2025
May 29th, 2025 at 8:29:26 AM
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This is cool! I'm thinking probably it's better to include the two-card 21 hands. Also, the average number of cards converges to an exact value as the number of decks increases to infinity. I am wondering what that value is.
May 29th, 2025 at 10:03:17 AM
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For infinite decks, I get these numbers:
Hit Soft 17:
Ace: 1.2446 (1,544,225,181,227 / 1,240,726,426,641)
2: 1.7264 (237,998,575,751 / 137,858,491,849)
3: 1.6030 (16,999,172,387 / 10,604,499,373)
4: 1.4834 (1,210,054,019 / 815,730,721)
5: 1.3671 (1,115,159,839 / 815,730,721)
6: 1.2621 (79,196,198 / 62,748,517)
7: 0.7914 (49,656,699 / 62,748,517)
8: 0.6714 (3,240,551 / 4,826,809)
9: 0.5599 (2,702,727 / 4,826,809)
10: 0.4859 (166,531 / 342,732)
Stand Soft 17:
Ace: 1.0115 (96,536,194,327 / 95,440,494,357)
2: 1.7051 (18,081,655,299 / 10,604,499,373)
3: 1.5838 (16,795,874,915 / 10,604,499,373)
4: 1.4656 (1,195,532,771 / 815,730,721)
5: 1.3585 (1,108,168,127 / 815,730,721)
6: 1.1507 (72,204,486 / 62,748,517)
7: 0.7914 (49,656,699 / 62,748,517)
8: 0.6714 (3,240,551 / 4,826,809)
9: 0.5599 (2,702,727 / 4,826,809)
10: 0.4859 (166,531 / 342,732)
Hit Soft 17:
Ace: 1.2446 (1,544,225,181,227 / 1,240,726,426,641)
2: 1.7264 (237,998,575,751 / 137,858,491,849)
3: 1.6030 (16,999,172,387 / 10,604,499,373)
4: 1.4834 (1,210,054,019 / 815,730,721)
5: 1.3671 (1,115,159,839 / 815,730,721)
6: 1.2621 (79,196,198 / 62,748,517)
7: 0.7914 (49,656,699 / 62,748,517)
8: 0.6714 (3,240,551 / 4,826,809)
9: 0.5599 (2,702,727 / 4,826,809)
10: 0.4859 (166,531 / 342,732)
Stand Soft 17:
Ace: 1.0115 (96,536,194,327 / 95,440,494,357)
2: 1.7051 (18,081,655,299 / 10,604,499,373)
3: 1.5838 (16,795,874,915 / 10,604,499,373)
4: 1.4656 (1,195,532,771 / 815,730,721)
5: 1.3585 (1,108,168,127 / 815,730,721)
6: 1.1507 (72,204,486 / 62,748,517)
7: 0.7914 (49,656,699 / 62,748,517)
8: 0.6714 (3,240,551 / 4,826,809)
9: 0.5599 (2,702,727 / 4,826,809)
10: 0.4859 (166,531 / 342,732)
May 30th, 2025 at 4:46:08 AM
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The Ace is a tricky card because it is soft. If it is counted as a 1 half the time and 11 another half, then the average value of an Ace should be
(1+11) / 2 =6.
From this point of view, we can just treat the Ace as the 6 to calculate the number of cards in a hand.
However, based on the above results of yours, we see that the Ace actually is counted as an 11 slightly more often than as an 1. Is this correct?
(1+11) / 2 =6.
From this point of view, we can just treat the Ace as the 6 to calculate the number of cards in a hand.
However, based on the above results of yours, we see that the Ace actually is counted as an 11 slightly more often than as an 1. Is this correct?
May 30th, 2025 at 11:38:47 AM
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Quote: acesideThe Ace is a tricky card because it is soft. If it is counted as a 1 half the time and 11 another half, then the average value of an Ace should be
(1+11) / 2 =6.
From this point of view, we can just treat the Ace as the 6 to calculate the number of cards in a hand.
However, based on the above results of yours, we see that the Ace actually is counted as an 11 slightly more often than as an 1. Is this correct?
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I do not believe that your analyses of the number of cards in a hand will be correct at all by assigning the Ace an average value of 6 or 6.5.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
May 30th, 2025 at 12:28:48 PM
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Still, let me stretch this a little.
For Hit Soft 17, the average number of cards in a dealer hand is
(1.2446+1.7264+1.6030+1.4834+1.3671+1.2621+0.7914+0.6714+0.5599+0.4859x4)/13+2=2.973.
For Stand Soft 17, the average number of cards in a dealer hand is
(1.0115+1.7051+1.5838+1.4656+1.3585+1.1507+0.7914+0.6714 +0.5599+0.4859x4)/13+2=2.942.
These numbers seem larger than the simulation results, as I recall.
For Hit Soft 17, the average number of cards in a dealer hand is
(1.2446+1.7264+1.6030+1.4834+1.3671+1.2621+0.7914+0.6714+0.5599+0.4859x4)/13+2=2.973.
For Stand Soft 17, the average number of cards in a dealer hand is
(1.0115+1.7051+1.5838+1.4656+1.3585+1.1507+0.7914+0.6714 +0.5599+0.4859x4)/13+2=2.942.
These numbers seem larger than the simulation results, as I recall.
May 30th, 2025 at 12:41:08 PM
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Did you remember to take into account that, for H17, for upcard Ace, the number should be 2 + 1.2446 x 9/13, and for upcard 10, it should be 2 + 0.4859 x 4 x 12/13? My numbers ignored 2-card 21s, since the OP asked for how many cards were drawn, so I assumed (a) two cards were dealt, and (b) they were interested in a value that was being played, rather than a 2-card 21.
May 30th, 2025 at 12:59:55 PM
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One more question. When the dealer upcard is a 6 and the dealer hole card is an Ace in a Stand-Soft-17 game, do you count this as zero card to draw?
May 30th, 2025 at 2:20:32 PM
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Quote: acesideOne more question. When the dealer upcard is a 6 and the dealer hole card is an Ace in a Stand-Soft-17 game, do you count this as zero card to draw?
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Yes. Isn't that what "Stand on Soft 17" means - stand on all 17s?
