Calculating '21s' from a single deck....

Hi everyone!!

I need to know how many combinations of '21' there are with 2 and 3 card combinations using a single deck of cards AND what percentage that figure represents in comparison to all possible 2 and 3 card hands dealt. In other words, is the likelyhood of getting a 2 or 3 card '21' a 1 in 5; a 1 in 10 etc....?

Hope you can help!
CHEERS!

Marty.

I gotta ask, why you are posing such a weird question.

In the meantime, I'll remind members what I had posted, a few days ago, here.

Haha! It's not a weird question, it's an important question!! I'm sure someone can calculate the overall probability of getting any kind of 2 or 3 card '21' from a single deck.

:-)

Well, fortunately I've since discovered that there are 769 ways of making '21' from either 2 or 3 cards using a single deck, but I need to know what percentage that reflects - such as 1% or 10% of all hands etc....

Any help?

Mart.

I dont think this is the right place to post this but I am going to reply anyways. I have my bachelors degree in statistics from emporia state university. I very recently brought stanford wongs casino tournament strategy book. Because I was trying to think if the math for it was right for what I was thinking as far as a blackjack method that did not involve counting but consistently beat the house. I also bought blackjack software and tested it for 2,000,000,789 hands. My results for this came up stunning I developed what I thought is a a variation of deep stack level progression non going broke betting strategy based on the laws of probability over time. This produces a positive gain for the player and let me explain how it works by explaining what it is not.
We know the math behind martingale doesnt work because you eventually go broke and you are betting to get one unit. What if you took a betting progression strategy and gave yourself aq bankroll of 200 times that betting progression and never went beyond that level you would then becoming up with something that beats the math behind most games because you never progress beyond that barrier of that which you have 200 times the total amount of the progresssion.
In even money games like blackjack the odds of winning one hand without doubling and splittling is close to 48% the odds of winning one out of two hands in blackjack without doubling or splittling is 73% and the odds of winning one out of three hands without doing the same is 86%. So here are the progressions I ran in my computer giving each player 200 X the total amount of each progression to see if we could yield a positive gain running each trial a minimum of 2,000,000,789 hands to account for the proper adjustments, I did 6 progression levels of each 200 deep for the total amount of progression so the players would not go broke if they lost one progression each player resorts back to the minimum bet after winning one progression bet

For the 1x then 5 X progression 200 deep my profit results yielded a constant growth of 12.7% percent over the long term
For the 1X 3X 15X progression 200 deep my profit results yielded a constant growth rate of 11.3% over the long term in simulation
For the 1X 2X 6X 18X progression 200 deep my profit results yielded a constant growth rate of 10.4% over the long term
For the 1x2X 6x 18X 72X progression 200 deep my profit resutls yielded a constant growth rate 9.6% over the long term
For the 1x 2x 6X 18x 54X 216X progression 200 deep my profit results yielded a constant growth rate of 8.7%
For the 1x 2x 6x 18x 54x 162X 1000X 200 deep my profit results yielded a constant growth rate of 6.5%

These trial progressions were ran 2,000,000,789 hands for each progression again the only reason this worked was because each player was sitting 200 deep of the total progression that way if a progression lost the player did not go broke. Cost is broken down as the following this is assuming a 5 dollar min wager

Bankroll for player A 5 + 25 = 30 * 200 = 6000 dollars
Bankroll for player b 5 + 15 + 75 * 200 = 19000 dollars
Bankroll for player c 5 + 10+ 30 + 90 * 200 = 27000 dollars
Bankroll for Player d 5 + 10 + 30 + 90 + 360 * 200 = 99000 dollars
Bankroll for Player e 5 + 10 + 30 + 90 + 270 + 1080 * 200 297000 dollars
Bankroll for Player F 5 = 10 + 30 + 90 + 270 + 810 + 5000 * 200 1,245,000 dollars

The whole premise behind this is to make sure you have 200 X of the total progression. Then the math does it self. Without the 200 X the strategy is no different then a glamoured up martingale. Without the 200 x the math says that this strategy will fail eventually. I would like someone else to do the math for me on this as well because maybe I have overlooked something, or maybe 2,000,000,789 is not enough hands. Yes that is right over 2 billion hands. Anyways get back to me on this please thank u

Jeremy Noble

Hi Jeremy.

I wasn't so much looking for a betting system or a playing strategy, I just needed to understand how many ways there are to make '21' using three cards and view that figure as a fraction. I know there are 128 ways to make 'blackjack' from the 2652 different two-carded combinations you could be dealt - but taking it one stage further seems quite a difficult task. Unless I'm missing a basic maths calcluation..!!

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Not quite. The total amount of any 2-card hand from a single deck is 52 x 51 = 2652.

Number of times you will have an Ace followed by face card is 4 x 16 = 64.

Number of times you will have face card followed by an Ace is 16 x 4 = 64.

Total number of blackjacks is 64 + 64 = 128/2652.

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It's semantics gentlemen - the difference between the phrases "two card combinations," and "how two cards are dealt."

There are twice as many ways to deal two cards as there are two card combinations.

6 times as many ways to deal 3 cards as there are 3 card combinations.

And 24 times as many ways to deal 4 cards as there are 4 card combinations.

Shall I continue?

Quote: DJTeddyBear

"how two cards are dealt."

The term you're looking for is permutations. While a combination denotes a specific way of selecting a specified number of elements at a time from a larger number of elements, a permutation denotes a way of arranging that specified number of elements. For example, the number of different two-pair-only hands in five-card poker is given by saying that, out of the thirteen ranks, three will be selected, two of those ranks in turn each selecting two of the four suits and one of them selecting one. The two-pair-only hand - single combination - you end up holding is, say Qh, Qd, 5d, 5c, 7s. While you may be dealt these cards one at a time in any of 5! = 120 possible sequences, the specific cards selected from the deck remain the same.

LOL. I see the forum 'bug' has you. Sarcasm, MATE, is not appreciated. First of all, you stated ACE/KING and KING/ACE as being 1 not 2 - even though you just gave two. Probability maths needs to address whether the order is important or not, and in this case is it not important so therefore there are two ways to acheive the same outcome. Besides, this is an easy answer to find as its plastered all over the internet on math expert sites and gambling statistic sites - and they ALL claim 128/2652. But like you said, you're not a maths expert.

But then again, Im not an English expert either and I think it was pretty clear that I was asking about three card combinations overall.

But thanks anyway....

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Sometime ago, for a laugh and before seeing https://wizardofodds.com/games/three-card-blackjack/, I looked at the totals that were formed with three cards (assuming an Ace was 1 or 11) but ignoring the totals from the first two cards (if you had 20 you wouldn't really hit to get the three-card 21). For instance there are four ways to get 6 (being three 2s and deciding which suit to omit), since A23 would be considered as 16.
30 560
29 480
28 576
27 740
26 856
25 1040
24 1180
23 1384
22 1544
21 2012
20 1688
19 1640
18 1540
17 1448
16 1304
15 1172
14 984
13 828
12 620
11 200
10 136
9 92
8 48
7 24
6 4

Quote: IGC

I know there are 128 ways to make 'blackjack' from the 2652 different two-carded combinations you could be dealt - but taking it one stage further seems quite a difficult task.

Quote: Ibeatyouraces

There are only 1326, 2 card combos in a single deck, 64 of which make 21 or blackjack. You forgot to divide by 2.

Are you actually arguing if the answer should be 128/2652 or 64/1326 ? Personally, I think you are both wrong. The correct answer is 32/663.

Also acceptable is one in 20+23/32.

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