Expert: Scotto - 9/3/2008

Question
How to use combinatorial analysis to calculate Baccarat probabilities?
Baccarat is a casino game using 8 decks of standard poker cards.
When a round is finished, some cards are dealt and seen, cards combinations in shoe(remaining cards) changed, how to calculate the chances of Player wins, Banker wins, and Tie, using combinatorial analysis.
I would put the formula into my program.
Thanks!

Answer
Since there are 8 decks of cards, that would mean that there are 32 of each card and 104 of each suit.  The probability of getting any particular card can be determined by the number of cards that have been seen.  The only difference the unseen cards make is in the number left to draw.

To compute the probability, there are four different numbers needed for that particular card:
 N: total cards left
 M: total of what is being looked for
 n: number of cards drawn
 m: number of looked for cards being drawn

If an 8 of hearts was drawn, there are now only 31 eights and 103 hearts.  Do the same thing for each card drawn.

I'll give you two examples.  My examples will assume this is the first time, but you'll have to subtract off the cards.

1) Suppose we are looking for getting four hearts in five cards drawn.  N is 416 (the number left), M is 32 (the number of aces left), n is 5 (the number of cards drawn), and m is 4 (the number of aces drawn).  To compute the number of ways, take
(N-M choose n-m)(M choose m)/(N choose n).

The formula 'a choose b' means a!/(b!(a-b)!) where ! is a factorial.  1!=1, 2!=2, 3!=6, 4!=24, n!=n*(n-1)!
This can be simplified to a(a-1)(a-2)...(a-b+1)/b!.
A good way to compute this if b gets large is to divide each of the top terms by a bottom term, so that you have
(a/b)((a-1)/(b-1))((a-2)/(b-2))....((a-n+2)/2)((a-n+1)/1).
I have just put that many terms in to show how to compute it.

2) Supppose we are looking for getting 4 hearts in 6 cards drawn.  
N is 416 (the number left), M is 104 (the number of hearts left), n is 6 (the number of hearts drawn), and m is 4 (the number of hearts drawn.  This would be (312 choose 2)(104 choose 4)/(416 choose 6).

To compute this, multiply the following:
(312/2)(6/416), (311/1)(5/415), (104/4)(4/414), (103/3)(3/413),(102/2)(2/413), and (101/1)(1/413).  Note that this includes all of terms needed.  A good reason to do it this way is the numbers quickly get to large.  For example, 14!=87,178,291,200, which is the highest my computer can express as an integer and
170!=7.25741561530799E+306 is the biggest factorial it can show.  Multiplying by 171 gives me #NUM! (undefined).

This is for only one combination of cards drawn.  To compute the odds of beating the banker, you would need to find every possible hand possible and do this combination for each (which seems like to much to me), then subtract this from one.

There may be an easier way, but I've not found it yet.