Flopping straight draws with a pocket connector: A tricky Texas Hold’em calculation

One of the more difficult calculations you’ll encounter in hold’em is the chance to get a straight and its friends the open straight and gutshot straight draw. In particular we are interested in the case when you have a connector and you’d like to know what are your straight chances on the flop. The full straight has only a 1.3% chance and the 8-out straight draws about 10.3%.

Made Straight

A connector is any two cards which have a successive rank. For example 5♥ 6♦ or Q♦ K♣ are connectors. Related to this is the suited connector, such as 9♥ T♥ which is both a connector and has a matching suit.

To simplify the calculation consider only those connectors in the middle: ones which a straight can be formed on either side of the two cards. The outer connectors require slightly modified calculations, but are essentially the same.

Given a connector there are four ways to form a straight (ignoring the suits). For example 67 can form the straight:

• 34567
• 45678
• 56789
• 6789T

That requires all three cards on the flop to be a specific rank. Each of these three cards can be any of the four suits. Thus there are (4*4*4)*4 = 256 possible flops resulting in a straight. Dividing by the 19600 total possible flops results in a 1.3% chance.

For the outer cases three cards are still needed, but the choice of those cards is limited. So for AK or A2 there are only 64 outcomes, KQ or 23 have 128, and QJ or 34 have 196.

Straight Draw

Since the odds of getting the straight on the flop itself are not so promising one can hope to get a straight draw. The two types of straight draw are the open ended straight draw, where a card on either end will complete it, and a gutshot straight draw where a card in the middle will complete it. It is decidedly more complicated to calculate these results.

Open Ended Straight Draw

The open ended straight draw, from a connector, has three possible forms on the flop. For example 45 can be an open straight draw as:

• 2345
• 3456
• 4567

The last card is irrelevant except that it cannot complete the straight.

Like before, each form has two cards which can be any of the four suits. In addition that final card can be any of the remaining cards except the ones completing the straight. There are four used cards and eight which complete the straight, so only 40 cards remain. That results in 3 * (4*4) * 40 = 1920 ways to get an open ended straight draw.

There is a slight problem however, kudos to those that notice it. The above inadvertently counts some hands twice, such as 2♥ 3♣ 4♠ 5♦ 2♣ and 2♣ 3♣ 4♠ 5♦ 2♥. Each pair combination has been permuted above resulting in 12 outcomes, but there are only 6 combinations. This 6 excess is repeated for six cards (two in each form), thus the 6 * 6 = 36 extra outcomes are subtracted from the total. The chance then is (1920-36) => 1884 / 19600 = 9.6% for an open ended straight draw.

Gutshot Straight Draw

The gutshot straight draw is one which has a card missing in the middle of the cards. There is a both a single gutshot and a double gutshot possible. The latter having an incomplete sequence in two directions, such as 4678T requiring a five or nine.

The single gutshot can appear in one of six forms. For example the cards 67 can form:

• 3467
• 3567
• 4678
• 5679
• 678T

This requires two cards and an extra remaining card which cannot complete the straight. That results in 6 * (4*4) * 44 = 4224 ways to get the gutshot straight draw. Again, the duplicate problem results in 12 *6 = 72 too many outcomes. The final result is thus 4224 – 72 => 4156 / 19600 = 21.2% chance of a gutshot.

The double gutshot is a bit more restrictive as it requires all three cards and has only two possible forms. For example the cards 67 can form:

• 35679
• 4678T

Unlike the two other straight draws this requires all three cards to be specific values. Each may however be a distinct suit. Thus there are 2 * (4*4*4) = 128 different possibilities, or a 0.65% chance of a double gutshot.

Note that the double gutshot is a subset of the single gutshot: every double gutshot is also a single gutshot.

Any 8-out Straight Draw

The double gutshot and the open ended straight draw are usually grouped into a single category. This is due to them both having eight outs to form a flush. Since these two classes don’t overlap the values can simply be added together to get their combined chance. Thus the chance to get an 8-out straight draw with a connector is (128+1884) / 19600 = 10.3%.