double-counting poker combinations

Quote: richodude

I'm trying to calculate the hand odds of 5-card stud with 4 and 5-card poker hands. To clarify all the normal poker hands with the addition of 4-card flushes and straights. For calculation purposes, a 4-card flush outranks a 4-card straight. Each of those hands lose to a three of a kind and beat a two pair. A math check is in dire need.

The question comes with calculating combos with a pair/ high card. With a pair in 5-card poker you would just calculate the odds of a pair with 3 singletons. With the possibility of 4-card hands, those 3 singletons could complete the poker hands. Here's what I have so far:

pair=13*COMBIN(4,2)*(COMBIN(12,3)-(44/13))*(4^3-2)+44*16

The explanation is 13 possible ranks for the pair picking out 2 of 4 suits for that pair. we pick 3 more ranks out of 12 remaining for the singletons (all standard so far) and I subtract the 44 stretches (2 given pair of 2's, 4 given a pair of 7's, etc.) over the 13 ranks. I then multiply by 4^3 for the possible suits of the singletons and subtract 2 for the cases where all 3 singletons match either paired card. I then add back the straight flush with a pair possibilities because I think I double-subtracted them?

high card=(COMBIN(13,5)-10-78)*(4^5-24)

choosing 5 singletons out of 13 ranks and subtracting the 10 5straight stretches and 78(I calculated) 4straight stretches multiplied by 4^5 for all the suits of the singletons minus 4 5flushes and 20 4flushes

4flush=4*(COMBIN(13,4)-11)*39-120

4 suits times picking 4 cards out of 13 ranks - 11 straight flush stretches * 39 non-suited kickers(5 card flush would outrank) - 120 5-card straights with 4 suited cards (I calculated)

4straight=9*(4^4-4)*40+2*(4^4-4)*44-176*10

9 4straights with 2 kicker ranks that would complete a 5straight (ie. not A,2,3,4 or J,Q,K,A) times 4 suits for the 4 cards - 4 sflushes times 40 kickers that wont improve the hand added to that 2 4straights of the same thing times 44 kickers that wont improve the hand. I then subtract the 4straight hands that are also 4flush hands (again that I calculated) since flush outranks straight

right now in my sheet I'm double-counting ~7k hand combinations and can't figure out where they're coming from. numbers where "I calculated" could easily be wrong. Thanks for any help
link to original post

Quote: gordonm888

Make separate categories out of 4 card straight plus a pair, 4 card flush plus a pair, and 4 card SF plus a pair.
link to original post

Quote: richodude

Commenting on Mental's calculations, I'm curious as to how your code got lower answers than me. The 4flush is definitely too low at only 3x more likely than a 5flush but the 4straight is ballpark. I like the video poker prediction but this will be a stud game of player vs. dealer. Hoping to get into game invention and looking at a class in LV in August where corporations can buy final projects (table game ideas).
link to original post

Quote: Mental

To be clear, I am not calculating the number of 4-card hands. I am calculating the number of 4-card hands that are not contained within a higher paying hand. An 5-card flush contains 5 different 4-card flushes, right? A 5-card straight contains two different 4-card straights.

I am calculating the 5-card stud probabilities for getting paid for a certain hand, because that is what you need for the EV. I did not look at your spoiler, but I suppose you are just getting the raw numbers without accounting for counterfeiting.

If I make the 4-straight pay more than the 5-flush and 5-straight I will get a larger number of 4-straights.
link to original post

Quote: richodude

Quote: gordonm888

Make separate categories out of 4 card straight plus a pair, 4 card flush plus a pair, and 4 card SF plus a pair.
link to original post

This bit is really helpful. I started again today with a fresh mind and think I've got it. The cells marked "(calculated)" are my answers

my sheet results

4sflush	2,032
4flush w/ 5straight	360
4flush w/ pair	34,320
4flush w/ kicker	102,600
4flush (theoretical)	134,888
4flush (calculated)	134,888
4straight w/ 4flush	1,320
4straight w/ 5straight	10,200
4straight w/ pair	33,264
4straight w/ kicker	89,592
4straight (theoretical)	121,536
4straight (calculated)	121,536

4sflush	=36*46+8*47
4flush w/ 5straight	=4*30*3
4flush w/ pair	=4*COMBIN(13,4)*12
4flush w/ kicker	=4*COMBIN(13,4)*36-360
4flush (theoretical)	=4*(COMBIN(13,4)*48) - B22-B23
4flush (calculated)	=sum(B24:B25) - B22
4straight w/ 4flush	=(11*4^4)*12/256*10
4straight w/ 5straight	=10*(4^5-4)
4straight w/ pair	=11*(4^4-4)*12
4straight w/ kicker	=11*(4^4-4)*36-B27
4straight (theoretical)	=11*(4^4-4)*48-B27-B26
4straight (calculated)	=SUM(B28,B29)-B26

I used google sheets on cells A22-B33 for this with the order slightly different. I had it
4sflush (red)
4flush w/ 5straight (yellow)
4flush w/ pair (green)
4flush w/ kicker (green)
4straight w/ 4flush (red)
4straight w/ 5straight (red)
4straight w/ pair (green)
4straight w/ kicker (green)
4flush (orange)
4straight (blue)
4flush (calculated) (orange)
4straight (calculated) (blue)
where a red cell got subtracted from each respective total cell, a yellow got subtracted only from the theoretical, and green was what was added in the calculated cells. I'm sure there's a better way to display this information but I couldn't be bothered finding how.

There is a special case where a 4-card straight with a pair where the paired card also forms a 4-card flush. It happens to be that every single scenario of this forms a 4 card straight flush which I already subtracted out.
These calculations help me with the whole hierarchy

all other hands

royal flush	4
straight flush	36
4 card royal	188
quads	624
4 card straight flush	1,844
full house	3,744
flush	5,108
straight	10,200
trips	54,912
4 card flush	134,888
4 card straight	121,536
two pair	123,552
pair	1,030,656
high card	1,111,668
total (theoretical)	2,598,960
total (calculated)	2,598,960

I'm somewhat confident in these numbers but now have some decisions about the game. Should 4-card flush still outrank straight and two pair? The difference is little and would be more intuitive for people without the numbers and knowledge of 5-card hands. Next, how would I add sections for ace high/king high/queen high etc. combinations? Normally it wouldn't be so bad but the 4-card combos really screw you.

Commenting on Mental's calculations, I'm curious as to how your code got lower answers than me. The 4flush is definitely too low at only 3x more likely than a 5flush but the 4straight is ballpark. I like the video poker prediction but this will be a stud game of player vs. dealer. Hoping to get into game invention and looking at a class in LV in August where corporations can buy final projects (table game ideas).
link to original post

Quote: Mental

Okay, I just looked at my code and saw a single character typo in the 4-flush code.

       Hand,     Pay,   #Flops
      Royal,  800.00,        4
  Str_Flush,   50.00,       36
      Quads,   25.00,      624
 Full_House,    9.00,     3744
      Flush,    6.00,     5108
   Straight,    4.00,    10200
      Trips,    3.00,    54912
     Flush4,    2.70,   110940
  Straight4,    2.30,    92208
   Two_Pair,    2.00,   123552
        JOB,    1.00,   323268
       Nada,    0.00,  1874364

       Hand,     Pay,   #Flops
      Royal,  800.00,        4
  Str_Flush,   50.00,       36
      Quads,   25.00,      624
 Full_House,    9.00,     3744
      Flush,    6.00,     5108
   Straight,    4.00,    10200
      Trips,    3.00,    54912
     Flush4,    2.70,   110940
  Straight4,    2.30,    92208
   Two_Pair,    2.00,   123552
        2OB,    1.00,  1047552
       Nada,    0.00,  1150080

Quote: Mental

I added in SF4 and RF4 hands because these eat up some of the 4-flush and 4-straight hands.

        Hand,     Pay,   #Flops
      Royal,  800.00,        4
  Str_Flush,   50.00,       36
        RF4,   40.00,      188
      Quads,   25.00,      624
        SF4,   15.00,     1844
 Full_House,    9.00,     3744
      Flush,    6.00,     4792
   Straight,    4.00,     9960
      Trips,    3.00,    54912
     Flush4,    2.70,   109464
  Straight4,    2.30,    92208
   Two_Pair,    2.00,   123552
        2OB,    1.00,  1047552
       Nada,    0.00,  1150080

So, we agree on everything except the straights and flushes.

I believe your flush number is wrong. You have flush = 5,108, but that is the total number of flushes that are not SF5 and RF5. Some of the Flushes are also also RF4 and SF4, so the number of plain 5-flushes has to be less than 5,108. Am I missing something about how you are defining and counting hands? A hand can only evaluate to one type, right?

Here are the flops for the traditional JOB VP hands.

       Hand,     Pay,   #Flops
      Royal,  800.00,        4
  Str_Flush,   50.00,       36
      Quads,   25.00,      624
 Full_House,    9.00,     3744
      Flush,    6.00,     5108
   Straight,    4.00,    10200
      Trips,    3.00,    54912
   Two_Pair,    2.00,   123552
        JOB,    1.00,   337920
       Nada,    0.00,  2062860

link to original post

    if temp_suit.count(0) == 4 or temp_suit.count(1) == 4 or temp_suit.count(2) == 4 or temp_suit.count(3) == 4:
        is_4flush = True 

Quote: richodude

My responses are usually quick. Why the delay? well I've been teaching myself to program of course! It's far from perfect, but I believe my logic for the 4-card hands is correct. Well, what did I get? Out of 2.6m random hands, ~95k of them were 4-card straights and ~89.5k were 4-card flushes. Along with ~1.16m high cards, ~1.05m pairs, ~54.8k trips and so on. Both of these 4-card hands still contain straight flush combos and I'm working that out but its 6:30 in the morning and I want to share something before I crash. That means your 4straight result is most likely correct! But how is my mathematical solution wrong? My brain already hurts and I don't wanna figure that out. That means then the actual number of 4flush combos should be closer to 86k, unless my code logic is undercounting something but it shouldn't with the lines

    if temp_suit.count(0) == 4 or temp_suit.count(1) == 4 or temp_suit.count(2) == 4 or temp_suit.count(3) == 4:
        is_4flush = True 

where suits have a value of 0-3 hearts-spades. The hand is ranked higher than 4straight so that's not undercounted. I'd take another look at how you're counting 4flushes. I'll definitely have another go tomorrow and use this code to build onto the full game. I could provide the code somewhere if you'd like

Also, yes. That 5108 flush count is a silly mistake from me
link to original post

void handInfo::setNada(int type)
{
  int k, int i0, i1, i2, i3, i4;
  if (type < 0) return; // isNada type was not found in pay table
  for (i0 = 0; i0 < 48; i0++) {
    for (i1 = i0 + 1; i1 < 49; i1++) {
      for (i2 = i1 + 1; i2 < 50; i2++) {
        for (i3 = i2 + 1; i3 < 51; i3++) {
          for (i4 = i3 + 1; i4 < 52; i4++) {
            k = indexHand(i0, i1, i2, i3, i4);
            tentative(k, type);
          }
        }
      }
    }
  }
}

Quote: Mental

Is this C++?

import itertools
combinations = itertools.combinations(range(0,52),5)
   for hand in combinations:
      rank_the_hand()