1. The Basics
  2. The Helpful Basics
  1. Total Combinations - In Holdem, you are a supplied with 7 cards, 2 in your hand and 5 community cards. With 7 cards, there are
    1. 52 choose 7
    2. =52*51*50*49*48*47*46/7/6/5/4/3/2
    3. =133784560 combinations
    4. So there are 133784560 possible combinations one can get using 7 cards.
  2. Royal Flush - Of the 133784560 combinations, there are
    1. 4 Royal Flush hands that requires 5 cards and 2 remaining unused cards
    2. 4 * 47 choose 2
    3. =4324
    4. So there are 4324 possible cominiations to get a Royal Flush
    5. The chances of you getting a Royal Flush is 4324/133784560 = 0.00003232062
  3. Straight Flush - To prevent overlaps in counting, we can break into cases
    1. Case A: seven cards straight flush
    2. By counting with finger, there are Ace-7 to 8-Ace, a total of 8 combinations in each suit
    3. Case B: exactly six cards straight flush
    4. There are 9 combinations times the one dangling card that cannot be adjacent
    5. The outside straights where the hand has an Ace-6 straight, the dangling card cannot be a 7. When the straight hand is a 9-Ace, the dangling card cannot be an 8
    6. So the calculation when the straight hand contains an Ace is 2*45=90
    7. The inside straights means the dangling card cannot be two cards.
    8. When the straight hand does not contian an Ace, is 7*44=308
    9. Case C: exactly five cards straight flush
    10. There are 10 combinations times the two dangling card that cannot be adjacent
    11. Using the same idea equates to 2*(46 choose 2)+8*(45 choose 2)=9990
    12. In total, we also need to multiply by the four suits, we get
    13. (8+90+308+9990) * 4 = 41584
    14. So there are 41584 combinations of straight flushes.
    15. The odds of getting a straight flush is 41584/133784560= 0.00031082809
    16. One thing to note is that all Royal Flushes are Straight Flushes. So that calculation includes Royal Flushes
  4. Four of a kind There are, by counting with your fingers, 13 four of kind hand using four cards. So in total, there are
    1. =13 * 48 choose 3
    2. =224848
    3. Odds: 224848/133784560=0.00168067226
  5. Full House - Breaking this into cases
    1. Case A: 2 triplets and one single
    2. (4 choose 3)*13*(4 choose 3) *12*11*4/2=54912
    3. Case B: 1 triplet, and 2 pairs
    4. (4 choose 3)*13*(4 choose 2)*12*(4 choose 2)*11/2 = 123552
    5. Case C:1 triplet, 1 pair and 2 singles
    6. (4 choose 3)*13*(4 choose 2)*12*11*4*10*4/2=3294720
    7. In total,
    8. 54912+123552+3294720=3473184
    9. Odds: 3473184/133784560 = 0.0259610227
  6. Flush - There are 13 choose 5 = 1287 combinations to get a five card flush in each suit.
    1. Total combinations of flushes in all four suits using five cards is
    2. 1287 * 4 = 5148
    3. One again, we need to break this into three cases
    4. Case A: all seven cards in the same suit
    5. =4 *(13 choose 7)=6864
    6. Case B: six cards in the same suit and one card in a different suit
    7. 4 *(13 choose 6) *39=267696
    8. Case C: five cards in the same suit and two cards in a different suit
    9. 4 *(13 choose 5)* (39 choose 2)=3814668
    10. In total
    11. 6864+267696+3814668=4089228
    12. Odds: 3950804/133784560 = 0.02953109088
    13. Do note that the above calculation include straight flush
  7. Straight - With straights, we have to remove flushes
    1. 7 distinct cards generate 4*4*4*4*4*4*4=16384 unique cards
    2. Of the 16384 uniques cards, there are exactly 4 seven card flushes, 4*3*7 six card flushes and 4*3*3*(7 choose 2) five card flushes
    3. 4+84+756
    4. =844
    5. To remove flushes from any 7 distinct cards would be 4*4*4*4*4*4*4-844=15540
    6. With 6 distinct cards where the seventh card is a matching pair, there would be 4 six card flushes and 4*3*6=72 five card flushes should the mathching pair not exist
    7. We also need to make sure that the match pair doesn't create a flush as well. So the matching pair will create a flush 4*3*5=60 times
    8. 4+72+60=136
    9. To remove flushes from any 6 distinct cards would be 4*4*4*4*4*4-136=3960
    10. We need to remove flushes as we calculate the straights
    11. Case A1: 7 card straight
    12. 8*(15540)=124320
    13. Case A2: 6 card straight with no pairs
    14. 2*((15540)*6)+7*((15540)*5)=730380
    15. Case A3: 5 card straight with no pairs
    16. 2*((15540)(7 choose 2))+8*(15540)*(6 choose 2))=2517480
    17. Case B1: 6 card straight with a pair matching the straight. We have to divide by 2 because the pairs generate overlap
    18. 2*((3960*(18)/2))+7*((3960*(18)/2))=320760
    19. Case B2: 5 card straight with exactly one pair
    20. 2*((3960*(18 choose 1)/2*(7 choose 1)))+8*((3960*(18 choose 1)/2*(6 choose 1)))=2209680
    21. Case C1: 5 card straight with a triple
    22. 10*((4*4*4*4*4*(5 choose 1)*(3 choose 2)/3))=51200
    23. Of 51200, there are 10*4*(5 choose 1)*(3 choose 2)*10=600 flushes
    24. 51200-600=50600 straights with triples
    25. Case C2: 5 card straight with two different rank cards matching the straight as pairs
    26. 10*(4*4*4*4*4(5 choose 2)*3*3/4)=230400
    27. Of the 230400, there are 10*4*(5 choose 2)*3*3=3600
    28. 230400-3600=226800
    29. In total,
    30. 124320+730380+2517480+320760+2209680+226800+50600=6180020
    31. odds: 6180020/133784560 = 0.04581560084
  8. Three of a kind - using the same calculations ideas...
    1. 6 461 620
    2. Odds: 6461620/133784560 = 0.04829869754
  9. Two Pairs - Once again, same idea
    1. 31 433 400
    2. Odds: 31433400/133784560 = 0.23495536405
  10. A Pair - ....
    1. 58 627 800
    2. Odds: 58627800/133784560 = 0.4382254574
  11. So now you know the poker odds of each hand. But knowing this, although is nice, is not really helpful. So, my next topic will be the helpful basics. If I have not written, it is coming.
If you want more explanation for counting odds, refer to Brian Alspach
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