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cnichols0021
Posted by cnichols0021
May 25, 2011

Black Jack Combinations Needing Help With The Math

So I've been trying to figure out the number of possible card combinations whose values will equal 21 from a single 52-card deck. I used the COMBIN() formula in Excel to figure out the total number of card combinations. The most cards that can be used to return a value of 21 is 11 (Ace, Ace, Ace, Ace, Two, Two, Two, Two, Three, Three, Three). Even though it would be played normally like this, it is possible to play the Aces as if they had a value of 1 and not 11. So the formula would be:
COMBIN(52,2)+COMBIN(52,3)+COMBIN(52,4)+COMBIN(52,5)+COMBIN(52,6)+COMBIN(52,7)+COMBIN(52+8)+COMBIN(52,9)+COMBIN(52,10)+COMBIN(52,11)= 80,812,402,801

But there is also the possibility that you could use the aces as if they had a value of 11. Does my formula take that into consideration?

Also, out of those 80-billion combinations, how could I figure out how many values equal 21?

Comments

cnichols0021
cnichols0021 May 27, 2011

For anyone who has read this post, I was able to figure it out. That formula is not correct since ACE's can be played two ways, as 1 or 11. For a single deck there are 128/3072 two card combinations for 21, 12216/165024 three card combinations, 304896/867728 four card combinations, 4616040/446509440 five card combinations....I havent got the numbers for 6-11 card combinations yet. I used a Microsoft Access query to run all of the possible combinations but once you get to six cards and above the query takes HOURS to return a result. I let it run all night last night and it crapped out on me this morning.



If you look at the percentages for each card combination, after 3 cards have been drawn and you havent got "21" you're chances for "21" drop considerably! 7.4% (3 cards) to 3.5% (4 cards) to 1.03% (5 cards)!!!



If I can get the results for 6 to 11 cards I will post. My hopes are to evaluate the strength of each card in the deck. I've read "How To Beat The Dealer" and "The World's Greatest Black Jack Book" and both of them mention the strength of cards. But I'm starting to see different results...