Five-Card Omaha: what a fifth hole card changes

Five-card Omaha is Pot-Limit Omaha with one more hole card. Nothing about the way you make a hand changes, and that is what makes the extra card easy to misjudge. This post counts what the fifth card adds, so the effects can be checked rather than guessed.

The rule does not move

In standard Omaha you must use exactly two of your hole cards and exactly three from the board. Dealing five cards instead of four does not let you use three, and it does not let you use one. You choose two cards from your hand, and the other three do nothing at showdown.

So the fifth card is not extra strength on the table. It is extra choice about which two cards to play, and it is a bigger set of cards that might happen to fit the board.

Ten pairs, not six

The number of two-card pairs you can pick is C(n,2):

• Four hole cards: C(4,2) = 4 × 3 ÷ 2 = 6 pairs.
• Five hole cards: C(5,2) = 5 × 4 ÷ 2 = 10 pairs.

The board contributes three cards from five, and C(5,3) = 10 ways to pick them. Every pair can meet every board triple, so at the river the number of different five-card hands you can build is:

• Four hole cards: 6 × 10 = 60 hands.
• Five hole cards: 10 × 10 = 100 hands.

That is 1.67 times as many hands to choose from, from a hand that is only 1.25 times as large. Pairs grow faster than cards, because each new card pairs with all the old ones: the fifth card adds 4 new pairs, one with each of the other four.

The starting hand is common more often

There are C(52,4) = 270,725 possible four-card hands and C(52,5) = 2,598,960 five-card hands. Counting the ones with certain features gives the figures below.

A pair in your hand. The hands with no pair at all use five different ranks out of 13 and any suit for each: C(13,5) × 4⁵ = 1,287 × 1,024 = 1,317,888. That is 50.7% of the 2,598,960, so 49.3% of five-card hands contain at least one pair. For four cards the same count is C(13,4) × 4⁴ = 715 × 256 = 183,040 of 270,725, which leaves 32.4% with a pair.

Two or more aces. Sum over the number of aces held. For four cards this is 2.57%, and for five cards 4.17%. Aces in hand are no stronger than before, but the reasons a pair of aces is weaker in Omaha apply here too: you still play only two of the cards, and the rest are dead weight.

Suits. Four cards can be a rainbow, with one card of each suit, in 13⁴ = 28,561 ways, which is 10.5% of hands. Five cards cannot. Five cards in four suits must repeat a suit, so every five-card hand has at least one suited pair, and the count of hands with three or more cards of one suit rises from 17.5% to 37.1%.

Danglers matter less, connection matters more

For straights, a pair of hole cards helps only if both ranks can sit in one straight, which means no more than four ranks apart. In the dangler count, J-T-9-3 had 3 working pairs out of 6, because the 3 is out of range of all three others.

Add a fifth card and the picture changes. Take J-T-9-8-3:

• The four connected cards give C(4,2) = 6 working pairs.
• The 3 still pairs with none of them, so it adds nothing.
• Result: 6 working pairs out of 10, against 3 out of 6 before.

The share of working pairs rises from 50% to 60%. Compare J-T-9-8-7, where all 10 pairs work: the 7 adds 4 working pairs, one with each of the other cards. A fifth card that does not connect adds none. The dangler is still dead weight, but it is one card in five rather than one in four, so it costs the hand less. The wrap draws covered earlier rest on the same idea: more working pairs, more ways to complete a straight.

What to take from the counts

1. You play exactly two hole cards in both games, so count pairs, not cards.
2. Ten pairs and 100 showdown hands mean more combinations of your cards can fit a board.
3. Strength still depends on the board and on how many of your ten pairs really work, so test each pair before you value the hand.
4. With five cards you always hold a suited pair, so the useful question is how high that suit runs and how many cards share it, not whether you have a suit at all.

The same arithmetic works for any card count: C(n,2) pairs, times C(5,3) = 10 board triples, gives the hands you can build.