Double-suited hands: what two suits add before the flop

Every Omaha starting-hand chart marks double-suited hands as better than the same ranks with one suit or none. The claim is easy to check, because suits are just counting. There are five ways four cards can be suited, each with an exact frequency, and each has an exact chance of making a flush. This post counts them.

The five suit patterns

A four-card hand is dealt from 52 cards in C(52,4) = 270,725 ways. Group those by how the suits split:

The five rows add back to 270,725. Single-suited is the commonest pattern by a distance. Double-suited is only third, at about one hand in 7.4, so most hands you are dealt will not have it.

The row for single-suited reads like this: choose the suit that appears twice (4 ways), choose its two ranks (78), choose which two of the other three suits supply the single cards (3), and give each a rank (13 × 13).

What each pattern does on the flop

For a flush you need two cards of a suit in your hand and three of that suit on the board, because exactly two hole cards play. Ranks do not matter for this count, only the suit pattern. After you hold four cards, 48 remain, and there are C(48,3) = 17,296 possible flops.

Take a single-suited hand, say two spades and two cards of other suits. Eleven spades are left in the deck.

• A flop of three spades makes the flush: C(11,3) = 165 flops, or 0.95%.
• A flop with exactly two spades gives a flush draw: C(11,2) × 37 = 2,035 flops, or 11.77%. The 37 is the non-spades left in the 48.

A double-suited hand has two such suits, and no flop can have two cards of each suit, since a flop only has three cards. So the cases never overlap, and the numbers simply double: 330 flops (1.91%) make a flush and 4,070 (23.53%) give a flush draw. Counting both, a double-suited hand flops a made flush or a flush draw 25.4% of the time. A single-suited hand does it 12.7% of the time and a rainbow hand never does, because no suit has two cards in the hand.

By the river

I dealt out every one of the C(48,5) = 1,712,304 five-card boards for each pattern and counted the boards that hand a flush (two hole cards of a suit plus three or more board cards of it). The totals are exact.

Double-suited hands make a flush by the river 245,124 times in 1,712,304, which is 14.3%. Single-suited hands make it 122,562 times, which is 7.2%, exactly half. That is the whole case for the second suit in one number: you are twice as likely to have a flush available.

Why a third card of the suit wastes a card

It might seem that more cards in one suit must be better. The table says otherwise. A hand with three spades makes a flush by the river 5.4% of the time, less than the 7.2% of a single-suited hand, and four spades only 3.9%.

There are two reasons, and both are plain counting.

1. You can only use two. The flush takes two cards from your hand and three from the board, so a third spade in your hand sits unused.
2. It removes a spade from the deck. With three in your hand, ten spades are left rather than eleven, and with four only nine. The board has fewer spades to find, and the flop numbers show it: 120 flops make the flush (against 165) and 1,710 give the draw (against 2,035).

The fourth card matters too. In a 3-1 hand it is a lone card in another suit that can never help a flush. In a 2-2 hand all four cards sit in a suit pair, so every card is working towards a flush.

What the numbers do not say

These are the chances of having a flush, not of winning with one. A flush draw is worth far less when it is not to the ace. A low suited pair makes a flush that a higher flush beats, as covered in second-best flush. A suit also counts for less when the ranks are poor: a double-suited 9-5-3-2 flushes as often as double-suited A-K-Q-J, and its flushes lose far more.

Suits are also only one part of a hand's value. Pairs are another, and they are why even aces are weaker in Omaha than in Hold'em.

A short rule for the table

1. Prefer 2-2 over 2-1-1 for the same ranks, since it doubles your flush chances.
2. Prefer 2-1-1 over 3-1 for the same ranks. The third card of the suit costs you a flush chance and gives nothing back.
3. Hold the high card of the suit. A suited ace is the nut flush draw, so a flush draw with one is worth more than one without it.

Omaha Poker grades your four-card hand before the flop, so you can deal a few hands and compare the grade of a double-suited hand with the same ranks in a rainbow. Its live hand insight shows your outs grouped by what they make, which makes it easy to see the flush draw you counted above.