Probability Derivations For Making Rank-based Hands in Omaha Hold 'em

Probability Derivations For Making Rank-based Hands In Omaha Hold 'em

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 boards on the flop, boards on the turn, and boards at the river. The probabilities for the boards in each table total 1.0.


Read more about Probability Derivations For Making Rank-based Hands In Omaha Hold 'em:  Derivations For Starting Hands With Four of A Kind, Derivations For Starting Hands With Three of A Kind, Derivations For Starting Hands With Two Pair, Derivations For Starting Hands With One Pair, Derivations For Starting Hands With No Pair, See Also

Famous quotes containing the words probability, making, hands, omaha and/or hold:

    Crushed to earth and rising again is an author’s gymnastic. Once he fails to struggle to his feet and grab his pen, he will contemplate a fact he should never permit himself to face: that in all probability books have been written, are being written, will be written, better than anything he has done, is doing, or will do.
    Fannie Hurst (1889–1968)

    The boredom of Sunday afternoon, which drove de Quincey to drink laudanum, also gave birth to surrealism: hours propitious for making bombs.
    Cyril Connolly (1903–1974)

    The Church disowned, the tower overthrown, the bells upturned, what have we to do
    But stand with empty hands and palms turned upwards
    In an age which advances progressively backwards?
    —T.S. (Thomas Stearns)

    The first dead man on Omaha Beach must be a sailor!
    Paddy Chayefsky (1923–1981)

    By many a legendary tale of violence and wrong, as well as by events which have passed before their eyes, these people have been taught to look upon white men with abhorrence.... I can sympathize with the spirit which prompts the Typee warrior to guard all the passes to his valley with the point of his levelled spear, and, standing upon the beach, with his back turned upon his green home, to hold at bay the intruding European.
    Herman Melville (1819–1891)