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Runner-runner outs in Texas hold ’em

Some outs for a hand require drawing an out on both the turn and the river—making two consecutive outs is called a runner-runner. Examples would be needing two cards to make a straight, flush, or three or four of a kind. Runner-runner outs can either draw from a common set of outs or from disjoint sets of outs. Two disjoint outs can either be conditional or independent events.

Common outs

Drawing to a flush is an example of drawing from a common set of outs. Both the turn and river need to be the same suit, so both outs are coming from a common set of outs—the set of remaining cards of the desired suit. After the flop, if x is the number of common outs, the probability P of drawing runner-runner outs is

P = (x/47) x ((x – 1)/46).

Since a flush would have 10 outs, the probability of a runner-runner flush draw is (10/47) x (9/46) = (90/2162) ~ 0.04163. Other examples of runner-runner draws from a common set of outs are drawing to three or four of a kind. When counting outs, it is convenient to convert runner-runner outs to “normal” outs. A runner-runner flush draw is about the equivalent of one “normal” out.

The following table shows the probability and odds of making a runner-runner from a common set of outs and the equivalent normal outs.

Likely drawing to Common outs Probability Odds Equivalent outs
Four of a kind (with pair)
Inside-only straight flush
2 0.00093 1080 : 1 0.02
Three of a kind (with no pair) 3 0.00278 359 : 1 0.07
4 0.00556 179 : 1 0.13
5 0.00925 107 : 1 0.22
Two pair or three of a kind (with no pair) 6 0.01388 71.1 : 1 0.33
7 0.01943 50.5 : 1 0.46
8 0.02590 37.6 : 1 0.61
9 0.03330 29.0 : 1 0.78
Flush 10 0.04163 23.0 : 1 0.98

Disjoint outs

Two outs are disjoint when there are no common cards between the set of cards needed for the first out and the set of cards needed for the second out. The outs are independent of each other if it does not matter which card comes first, and one card appearing does not affect the probability of the other card appearing except by changing the number of remaining cards; an example is drawing two cards to an inside straight. The outs are conditional on each other if the number of outs available for the second card depends on the first card; an example is drawing two cards to an outside straight.

After the flop, if x is the number of independent outs for one card and y is the number of outs for the second card, then the probability P of making the runner-runner is

P = (x/47) x (y/46) x 2 = (xy/1081).

For example, a player holding J♦ Q♦ after the flop 9♥ 5♣ 6♠ needs a 10 and either a K or 8 on the turn and river to make a straight. There are 4 10s and 8 kings and 8s, so the probability is (4×8)/1081 ~ 0.0296.

The probability of making a conditional runner-runner depends on the condition. For example, a player holding 9♥ 10♥ after the flop 8♦ 2♠ A♣ can make a straight with {J, Q}, {7, J} or {6, 7}. The number of outs for the second card is conditional on the first card—a Q or 6 (8 cards) on the first card leaves only 4 outs (J or 7, respectively) for the second card, while a J or 7 (8 cards) for the first card leaves 8 outs ({Q, 7} or {J, 6}, respectively) for the second card. The probability P of a runner-runner straight for this hand is calculated by the equation

P = ((8/47) x (4/46)) + ((8/47) x (8/46)) = 96/2162 ~ 0.0444.

The following table shows the probability and odds of making a runner-runner from a disjoint set of outs for common situations and the equivalent normal outs.

Drawing to Probability Odds Equivalent outs
Outside straight 0.04440 21.5 : 1 1.04
Inside+outside straight 0.02960 32.8 : 1 0.70
Inside-only straight 0.01480 66.6 : 1 0.35
Outside straight flush 0.00278 359 : 1 0.07
Inside+outside straight flush 0.00185 540 : 1 0.04

The preceding table assumes the following definitions.

Outside straight and straight flush
Drawing to a sequence of three cards of consecutive rank from 3-4-5 to 10-J-Q where two cards can be added to either end of the sequence to make a straight or straight flush.
Inside+outside straight and straight flush
Drawing to a straight or straight flush where one required rank can be combined with one of two other ranks to make the hand. This includes sequences like 5-7-8 which requires a 6 plus either a 4 or 9 as well as the sequences J-Q-K, which requires a 10 plus either a 9 or A, and 2-3-4 which requires a 5 plus either an A or 6.
Inside-only straight and straight flush
Drawing to a straight or straight flush where there are only two ranks that make the hand. This includes hands such as 5-7-9 which requires a 6 and an 8 as well as A-2-3 which requires a 4 and a 5.

Compound outs

The strongest runner-runner probabilities lie with hands that are drawing to multiple hands with different runner-runner combinations. These include hands that can make a straight, flush or straight flush, as well as four of a kind or a full house. Calculating these probabilities requires adding the compound probabilities for the various outs, taking care to account for any shared hands. For example, if Ps is the probability of a runner-runner straight, Pf is the probability of a runner-runner flush, and Psf is the probability of a runner-runner straight flush, then the compound probability P of getting one of these hands is

P = Ps + PfPsf.

The probability of the straight flush is subtracted from the total because it is already included in both the probability of a straight and the probability of a flush, so it has been added twice and must therefore be subtracted from the compound outs of a straight or flush.

The following table gives the compound probability and odds of making a runner-runner for common situations and the equivalent normal outs.

Drawing to Probability Odds Equivalent outs
Flush, outside straight or straight flush 0.08326 11.0 : 1 1.98
Flush, inside+outside straight or straight flush 0.06938 13.4 : 1 1.65
Flush, inside-only straight or straight flush 0.05550 17.0 : 1 1.30

Some hands have even more runner-runner chances to improve. For example, holding the hand J♠ Q♠ after a flop of 10♠ J♥ 7♦ there are several runner-runner hands to make at least a straight. The hand can get two cards from the common outs of {J, Q} (5 cards) to make a full house or four of a kind, can get a J (2 cards) plus either a 7 or 10 (6 cards) to make a full house from these independent disjoint outs, and is drawing to the compound outs of a flush, outside straight or straight flush. The hand can also make {7, 7} or {10, 10} (each drawing from 3 common outs) to make a full house, although this will make four of a kind for anyone holding the remaining 7 or 10 or a bigger full house for anyone holding an overpair. Working from the probabilities from the previous tables and equations, the probability P of making one of these runner-runner hands is a compound probability

P = 0.08326 + 0.00925 + (2×6)/1081 + (0.00278 x 2) ~ 0.1092

and odds of 8.16 : 1 for the equivalent of 2.59 normal outs. Almost all of these runner-runners give a winning hand against an opponent who had flopped a straight holding 8, 9, but only some give a winning hand against A♠ 2♠ (this hand makes bigger flushes when a flush is hit) or against K♣ Q♦ (this hand makes bigger straights when a straight is hit with 8 9). When counting outs, it is necessary to adjust for which outs are likely to give a winning hand—this is where the skill in poker becomes more important than being able to calculate the probabilities.

Notes

1. ^ In the example, if the opponent is holding either 8♥ 9♥ or 8♦ 9♦, then the opponent wins with a flush if the player makes a straight using two hearts or two diamonds, respectively. If the opponent is holding 8♦ 9♦, then the opponent wins with a straight flush if the player makes a full house with 10♦ J♦.