The following analysis will give you an idea of the probability of being dealt a certain poker hand. Consider a standard deck of cards (2, 3, ..., 10, J, Q, K, A; ♣, ♦, ♥, ♠). The number of different hands of five cards out of 52 (=number of possible combinations) is equal to:
                 (52)(51)(50)(49)(48)
          C552 = -------------------- = 2 598 960 hands
                    (1)(2)(3)(4)(5)
  • A Royal Flush requires 10-to-Ace in one suit. There are 4 suits, so there are 4 ways of making a Royal Flush.
  • A Straight Flush requires five cards in sequence of the same suit. There are ten straights in each suit, considering that the Ace can be either low or high: A, 2, 3, 4, 5 through 10, J, Q, K, A. Since there are 4 suits, this leads to 40 combinations. Subtract the 4 Royal Flushes results in 36 ways of making a Straight Flush.
  • A Four of a Kind requires four cards of the same denomination, and one other card. There are 13 denominations, with (12 x 4 =) 48 non- matching cards. In total, there are 13 x 48 = 624 ways of making a Four of a Kind.
  • A Full House consists of 3 cards with one denomination, and 2 of another. There are 4 ways to get three cards of any nomination:

    ♣ ♦ ♥
    ♦ ♥ ♠
    ♣ ♥ ♠
    ♣ ♦ ♠

    There are 13 denominations resulting in 4 x 13 = 52 ways to get three cards of one denomination. The pair must be in another denomination. There are 12 possible denominations, and 6 ways of pairing the denominations:

    ♣ ♦
    ♣ ♥
    ♣ ♠
    ♦ ♥
    ♦ ♠
    ♥ ♠

    Thus, the pair can be obtained in 6 x 12 = 72 ways. In total, there are 52 x 72 = 3 744 ways of making a Full House

  • A Flush consists of five cards in the same suit. Since there are four suits, there are:
                           (13)(12)(11)(10)(9)
              4*C513 = 4 * ------------------- = 5 148 combinations
                             (1)(2)(3)(4)(5)
    
    Subtract the 40 combinations that form a Royal or Straight Flush. This results in 5 108 ways to obtain a Flush
  • There are ten possible Straights, each with 4 x 4 x 4 x 4 x 4 ways of occurring. This equals 10 240 ways. Subtract the 40 combinations that a Straight Flush. This results in 10 200 ways to obtain a Straight.
  • Three of a Kind requires one of four ways of getting three cards of the same rank. Any one of (4 x 12 =) 48 cards will have a different rank from the triple. Any one of (4 x 11=) 44 cards will have a different rank from the triple and the other non-matching card. There are 48 x 44/2 = non-matching pairs that will fill the hand (order doesn't matter e.g. 9 Q is similar to Q 9). Thus, there are 4 x 13 x 48 x 44/2 = 54 912 ways to get a Three of a Kind.
  • Similarly, Two Pair consists of
              C24 x C24 x C213 x 44 = 6 x 6 x 13 x 12 x 44/2 = 123 552 combinations
    
  • One pair consists of
              13 x C24 x 48 x 44 x 40 / (3 x 2 x 1) = 1 098 240 combinations
    
  • There are 1 302 400 remaining deals that have no value (High Card).

Dividing each number by the total number of hands (2 598 960) yields the probabilities for each poker hand for the initial deal:

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Hand            Probability       %
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Royal Flush     0.000002         0.0002
Straight Flush  0.000014         0.0014
Four of a Kind  0.00024          0.0240
Full House      0.00144          0.144
Flush           0.00197          0.197
Straight        0.00392          0.392
Three of a Kind 0.02113          2.113
Two Pairs       0.04754          4.754
One Pair        0.42257         42.257
High Card       0.50112         50.112
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