Probability derivations for making rank-based hands in Omaha hold 'em

SeePoker probability (Omaha)#Making hands based on rank typefor the probabilities derived from these equations.

The probability derivations for starting hands making four of a kind, a full house, three of a kind, two pair, one pair and no pair in Omaha hold 'em are separate for each of the starting hand rank types.

The derivations require identifying the individual cases that yield each possible hand and are sometimes rather detailed, so it is useful to use a notation to indicate the shape of the board for each case. The rank type of the hand is shown using upper case letters to indicate ranks. The ranks on the board are indicated using upper case letters for matches with the starting hand and lower case letters to indicate ranks that don't match the starting hand. So the rank type XXYZ is any hand with a pair of X with two additional ranks Y and Z and the board XYr represents a flop that contains one X, one of the non-paired ranks Y and one other rank r. Note that since Y and Z have an identical relationship to the starting hand—each represents an unpaired rank—XYr and XZr represent the same set of boards and are interchangeable, so derivations for this hand choose one of the two choices represented by Y. In addition to the upper and lower case letters, * is used to represent any rank not already represented on the board, and ? is used to represent any rank not already represented on the board and not included in the starting hand. So for the rank type XXYZ, the board XX* represents a flop that contains two Xs and any other rank (including Y and Z), but X?? is any flop that contains an X and any two cards of a rank other than X, Y or Z, and rrr?? is any board on the river that contains three cards of rank r and any two cards of ranks other than X, Y, Z or r.

Each table shows all of the boards that can make each hand and the derivation for the combinations for that board. Probabilities are determined by dividing the number of combinations for each hand by the $\begin{matrix} {48 \choose 3} = 17,296 \end{matrix}$ boards on the flop, $\begin{matrix} {48 \choose 4} = 194,580 \end{matrix}$ boards on the turn, and $\begin{matrix} {48 \choose 5} = 1,712,304 \end{matrix}$ boards at the river. The probabilities for the boards in each table total 1.0.

Derivations for starting hands with four of a kind

Starting hands with four of a kind (XXXX) can only improve to a full house or two pair. To make a full house, this hand needs to have two or three cards of the same rank appear on the board. To make two pair, another pair on the board is needed. Of course, any other hand holding a pair also makes at least a full house or two with either of these boards. The following table shows the derivations for making a full house, two pair or one pair when holding four of a kind.