Flop Combinatorics

    • VorpalF2F
      VorpalF2F
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      Joined: 02.09.2010 Posts: 8,907
      I noticed last night that I was seeing a huge number of two-tone flops

      I had no idea what the real number should be, nor could I find it on line.

      I therefore decided to figure it out.

      I turns out that there are 19,600 possible flops. Order of cards is disregarded -- ie, :Ad: :Ts: :8h: is the same as :8h: :Ad: :Ts:

      I used Excel's combin(number,number_chosen) function for this, and in this case that would be C(50,3)
      50 because you hold two cards, so only 50 are unknown.
      It turns out that the percentages are slightly different if you hold an offsuit hand compared to the percentage when you hold a suited hand.

      The results:
      (hint: if this table looks really ugly, widen your browser to the mas)
                    Monotone    Rainbow    TwoSuited    Total         Ratio  
        Offsuit:        1012       7800        10788    19600    1:7.7:10.6  
         Suited:        1023       7774        10803    19600    1:7.6:10.5  
        Offsuit:          5%        40%          55%                         
         Suited:          5%        40%          55%                         


      Following this I checked my session, and I had 4 more 2-suited flops that the math would predict.
      I'm either *extremely* perceptive or it is just another example of seeing patterns where there are none.

      So it goes.

      If anyone is interested in the math, let me know and I'll post it.

      Peace,
      VS
  • 6 replies
    • GoOnCal1
      GoOnCal1
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      Joined: 22.01.2015 Posts: 581
      Originally posted by VorpalF2F

      I'm either *extremely* perceptive or it is just another example of seeing patterns where there are none.

      VS
      Apparently seeing patterns is an evolutionary essential trait, while seeing erroneous patterns hardly affects our survival, but missing a tiger in the tall grass only once is instant gene pool removal.
      :heart:
    • VorpalF2F
      VorpalF2F
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      Joined: 02.09.2010 Posts: 8,907
      Hi, GoOnCal1,
      Yes, that's what I was referring to.

      It happens a lot to poker players. That is how they develop "lucky" hands, and "AQo always loses"

      It happens at an unconscious (OK, subconscious :coolface: ) level with a great negative effect.
      If (for example) you lose the first several times you open-raise suited connectors, you may without thinking begin to remove them from your range.

      Have a great weekend,
      VS
    • Lazza61
      Lazza61
      Headadmin
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      Joined: 23.03.2011 Posts: 9,209
      Originally posted by VorpalF2F

      AQo always loses

      Possibly explains why Doyle Brunson almost always folds it.
    • VorpalF2F
      VorpalF2F
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      Joined: 02.09.2010 Posts: 8,907
      That is precisely who I was thinking of
    • VorpalF2F
      VorpalF2F
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      Joined: 02.09.2010 Posts: 8,907
      I flopped quads twice last week.
      Both times, my pocket pair hit the same pair on the flop.

      I'll admit that I found the math daunting, so I did what any other lazy person would do:
      I just counted 'em.
      I used a Python script to generate all the possible flops, then went through the list and built the following tables.

      I'm pretty sure this won't help anyone increase their winrate, but it is interesting...

      Suited hole cards
                    0 Pair    1 Pair    Trips    Total  
        Monotone      1023         0        0     1023  
         Rainbow      6072      1656       46     7774  
         TwoSuit      9141      1662        0    10803  
           Total     16236      3318       46    19600  


      Offsuit hole cards
                    0 Pair    1 Pair    Trips    Total  
        Monotone      1012         0        0     1012  
         Rainbow      6094      1660       46     7800  
         TwoSuit      9130      1658        0    10788  
           Total     16236      3318       46    19600  

      The number for trips looked odd, but if you do not have a pocket pair, then there are 11 ranks with 4 combos of trips, and two with 1 each.

      If you hold a pocket pair, then there are 12 ranks with 4 combos of trips each, and one rank with none, for 48.

      This made me realize that all of the above are wrong if you have an offsuit pocket pair.
      They're very wrong if you hold a suited pocket pair, actually.

      If you do hold a pocket pair, then of the 50 unknown cards, two match your pair, leaving 48. Each of those 48 can go with your pocket pair 1 way, so you any time you hold a pocket pair, you have a 48/19600 or about 0.24% chance of hitting quads.

      So yay me for doing it twice in about 5K hands.

      Best of luck,
      VS
    • VorpalF2F
      VorpalF2F
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      Joined: 02.09.2010 Posts: 8,907
      OK, I ran the numbers for flop combinations when you hold a pocket pair.

      Pocket pair
                    0 Pair    1 Pair    Trips    Total  
        Monotone      1012         0        0     1012  
         Rainbow      6072      1680       48     7800  
         TwoSuit      9108      1680        0    10788  
           Total     16192      3360       48    19600  


      Cheers,
      VS