Math behind the BJ charts

Hello

I tried to search the answer for my question with the search function, but I did not succeed so I decided to create this thread. If this should be answered already forgive me, i'm sorry.
I'm wondering how the BJ charts are calculated.
I understand that when you have a 10 and a 6 and the dealer has a 8 the chance when you bust after you hit in a 52card deck is (16+4+3+4+3)/49 so 61%. So first thought would be it's better to stand, but that's not true because even if it is -EV, it could still be better than standing where the EV would be more negative. So if I try to calculate this for standing would that mean the following:
The chance that the dealer wins directly according to the stand on 17 or higher is (4+16+4)/49 is 49% then there all the other combinations like if the second card would be an 8, then there is the chance that the dealer wins 3/49*(4+4+4+4+5)/48=3%, this makes the subtotal 52% and so on if there is a 7,6,5,4,3,2,A as second card. There is even the chance that the dealercards turn out to be 822AAAA2.
So first is my reasoning correct and if so is there an easy formula to calculate these things?

Regards

Joris

The math has already been worked out for every possible combination, so you can easily identify what the best move is for any combination of two cards plus a dealers.
Are you more interested in how to get the answer or in the answer itself? Most people just want to learn basic Strategy, not the math behind each hand.
The original Basic Strategy was done by a couple of air force guys using one of the most powerful computers available at the time. I think your phone is more powerful than that old computer.

Quote: joris1992

I understand that when you have a 10 and a 6 and the dealer has a 8 the chance when you bust after you hit in a 52card deck is (16+4+3+4+3)/49 so 61%. So first thought would be it's better to stand, but that's not true because even if it is -EV, it could still be better than standing where the EV would be more negative. So if I try to calculate this for standing would that mean the following:
The chance that the dealer wins directly according to the stand on 17 or higher is (4+16+4)/49 is 49% then there all the other combinations like if the second card would be an 8, then there is the chance that the dealer wins 3/49*(4+4+4+4+5)/48=3%, this makes the subtotal 52% and so on if there is a 7,6,5,4,3,2,A as second card. There is even the chance that the dealercards turn out to be 822AAAA2.
So first is my reasoning correct and if so is there an easy formula to calculate these things?

It's a little hard to work out your reasoning, but I think you're on the right track. However, there's no easy way because of the recursions involved.

Here's the short version of how I did it:
It has to be done for each of the ten possible values of dealer's up card separately.
First, I calculated the probability of the dealer getting 17, 18, 19, 20, 21, or busting for that particular up card value, ignoring two-card 21s.
Next, I calculated the probability of winning by standing with a 21 (i.e. you win if the dealer has anything besides 21, and push with a 21) against hitting with a 21 (i.e. you always bust).
Then, I calculated the probability of winning by standing with a 20 (win with dealer 17-19 or bust, push with 20, lose with 21) against hitting with 20 (this is 1/13 x the probability of winning with 21 + 12/13 x losing by busting; this is why the probability of winning with 21 had to be calculated first).
Continue doing this all the way down to 3.
You then check the soft hands - for example, if you hit soft 18, it's 1/13 x (probability of winning with 19) + 1/13 x (soft 20) + 1/13 x (soft 21) + 1/13 x (hard 12) + 1/13 x (hard 13) + ... + 4/13 x (hard 18). Again, since lower numbers can have higher soft hands, you have to start with soft 20 ("soft 21" is "always stand") and work down to soft 12.
Then, you check the probability of winning if you double a particular hand against the already calculated probability of hitting or standing with that hand, to see if doubling is better, remembering to take into account that the bet on the doubled hand is now doubled.

What you've described above is how to work it out using infinite decks. This is a good approximation towards a reasonable strategy. It assumes that if the dealer has 16, regardless of how it's made up, the chances of an Ace are 1/13, a 2 are 1/13 etc. Using this method you can create a table of the chances of getting to 21, 20, 19, 18, 17 and Bust from any given [Dealer's] starting total. Further calculation gives you the EV for the player standing on various totals, from which you can work out the EVs for hitting etc.

The long correct method, which I've only done for a few specific cases, is to look at what happens if you stand on your 10+6 vs 8, running through all the possible hands the dealer could make given the cards already known. This eventually gives you the chances of winning (or EV) if you stand. Then you look at 10+6+A and repeat the idea, similarly 10+6+2, 10+6+3, 10+6+4, 10+6+5. This gives five chances of winning (or EV) for each of your hit possibilities which don't bust. Factor these, together with the bust cases where you lose, by the chances of you drawing A, 2, 3 etc.

It should be possible to do this in a spreadsheet for the 10+6 vs 8, and if you generalise the code correctly for any card combination of the player's 16 and any number of decks vs dealer's 8. The iteration comes from needing to work out 10+6+A, 10+6+2...etc. so then you know what to do with 10+6. However, as an example, for 10+5 you would also have to work out what to do with 10+5+A first.

I'm guessing a full calculation keeps track of every (reasonable) possible player hand and which ones can be created by being hit from which others. Then for each upcard working out all the possible dealer hands and the Stand EVs. Finally starting at 21 and working backwards iteratng to work out for any given player hand whether to hit or stand and adjusting the EVs accordingly. Similarly doubling can be worked out and then splitting (unless you really want to get into catering for the fact you have split and know which cards are in your other hand(s) ).

Happy coding

Quote: joris1992

Hello

I tried to search the answer for my question with the search function, but I did not succeed so I decided to create this thread. If this should be answered already forgive me, i'm sorry.
I'm wondering how the BJ charts are calculated.

Hi Joris and welcome to the forum.
Here's something towards explaining that...

https://www.youtube.com/watch?v=jCF-Btu5ZCk

Quote: joris1992

So first is my reasoning correct and if so is there an easy formula to calculate these things?

Regards

Joris

depends on what you want from the data.
Infinite decks calculation is easy (in a spreadsheet) and even The Wizard has a video (or 2) on how to actually do it in Excel.
Composition Dependent Combinatorial Analyzer requires good computer code
here is a site that offers results
http://www.bjstrat.net/cgi-bin/cdca.cgi

their program for a Windows computer works well but I have yet to study the code for it.
your example returns this

****Blackjack Combinatorial Analyzer v1.60***
Player's Hand: 6 10
Dealer's Hand: 8
Rules: S17, Dealer Peaks
Deck Composition: Ace(4) Two(4) Three(4) Four(4) Five(4) Six(4) Seven(4) Eight(4) Nine(4) Ten(16)
*****
Results for Standing
p-1	p0	p+1	p+1.5	EV(units)	SD(units)	DI(EV/SD)
0.7635032619	0	0.2364967381	0	-0.5270065239	0.849861238	-0.6201089076
Results for Hitting
p-1	p0	p+1	EV(units)	SD(units)	DI(EV/SD)
0.6811683166	0.06248591671	0.2563457667	-0.4248225499	0.8700803896	-0.4882566657
Results for Doubling
p-2	p0	p+2	EV(units)	SD(units)	DI(EV/SD)
0.6811683166	0.06248591671	0.2563457667	-0.8496450999	1.740160779	-0.4882566657

If Blackjack math was so easy, many would have already done it hundreds of years ago, but not one did until a few started to use computers to figure out all the possible outcomes.
Good for them

Quote: 7craps

....If Blackjack math was so easy, many would have already done it hundreds of years ago, but not one did until a few started to use computers to figure out all the possible outcomes. Good for them

I'm slightly surprised people couldn't work out an infinite decks strategy; but to realise the effect of card removal required more thought.

Back in the 80s before I knew much about BlackJack I worked Stand/Hit/Split/Double strategy using infinite deck analysis, a calculator and a large pile of [mainframe] computer printouts for paper. I could only keep probabilities to 9 digits but it came up with the correct strategy and a clear understanding of the decisions - I remember working up from 2,3...7,8,9... I was surprised NOT to split 9's vs 7 (but could see the logic). Nowadays (ignoring re-splits) you can do it on a spreadsheet fairly quickly with a little thought.