Joined: Apr 10, 2013
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August 16th, 2013 at 8:36:08 AM permalink
Card removal and No Limit Hold'em ?

Hello, I'm still a relative beginner at counting combos and combinatorics, and I've been stumped lately on some math that I can't figure out. For some background on my situation, I know that: In No Limit Hold'em (NLHE), there are a total of 169 hands (1326 combos). Of those 169 hands; 13 hands are pairs (78 combos), and 78 hands are Non-Pairs (1248 combos). The Non-Pair hands break into the two groups of Off-suited hands and Suited hands. There are 78 Off-suited hands (936 combos) and 78 Suited hands (312 combos). As far as individual starting hands go; there are 6 combos per Pair hand, 16 combos per Non-Pair hand, 12 combos per Off-suited hand, and 4 combos per Suited hand. I am starting to try my hand at the math that shows how card removal (dead cards) affect the number of combos for a particular hand type. Pairs are easy; for 0, 1, 2, and 3 dead cards, there are 6, 3, 1, and 0 combos possible.

Main Question vvv

Where I'm starting to doubt my results is when calculating how the number of dead cards, from 2 - 6 decreases the amount of combos possible for Non-Pair, Off-suited, and Suited hands. So, I wonder if someone who knows about this topic can verify my results, or give correct answers so that I can go back and rethink how I'm going about calculating them? My results are shown below. Across the top, from left to right is the number of dead cards and then underneath there are three rows showing the # of possible combos for each of the 3 hand-types (Non-pairs AKx, Off-suited JTo, and Suited 98s):

# Dead
cards > 0, 1, 2, 3, 4, 5, 6.

combos 16, 12, 9, 5, 4, 2, 0.

combos 12, 9, 7, 4, 3, 2, 0.

combos 4, 3, 2, 1, 1, 0, 0.

The results are in whole numbers, so that they are more practical to use and apply in-game and when I'm seeing how ranges are affect by dead cards during a hand review. I'm wanting to know if they are accurate enough to use both during a live game, and also when I'm studying situations away from the table to get an idea of how I played a particular hand (good or bad)?

If the above "# combos" figures for the AKx hand are accurate enough, I may go back and use the more precise figures (to 3 decimal places) for the AKx hand. Then, I'd divide up the combos proportionally for the JTo and 98s hands (splitting the AKx hand into 3:1). I'd do this to see if it made a big enough difference to change the existing results for JTo and 98s. Once those two hands were sorted out into proper whole numbers, I could go back and round off the AKx figures to a whole number. Are the results above good enough to use for most situations that you know of? Thanks in advance for the help on this!

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