Counting Outs

[BB] Joey the Juvenile has been calling too much, and not on the phone. His dad, Benny the Book explained, “When you’re behind in a hand, you can’t just call every bet because you might improve to a better hand. You need to know what your chances are, and the first thing is to count how many cards will improve your hand enough to beat your opponent’s. These are called your outs:”

Draw Outs Notes
Flush Draw 9 Outs 13 – 4 = 9
Straight Draw 8 Outs 2 * 4 = 8
2 Overcards 6 Outs 2 * 3 = 6
Inside Straight Draw 4 Outs
1 Overcard 3 Outs

[BB] “If you have a combination draw, such as a straight flush draw, add the outs then subtract any duplicated outs. For example, if you’re pretty sure your opponent has a pair of Aces1 and you’re holding Q♥J♥ on a 10♥9♥2♣ board, you have 9 + 8 – 2 = 15 outs (the K♥ and 8♥, which actually give you a straight flush, being the two duplicated outs).”

[BB] “Once you know how many outs you have, you can convert to a percentage or odds using this chart:”

Outs Turn Odds River Odds Turn+River Odds
22 Outs 1.1-to-1 (46.81%) 1.1-to-1 (47.83%) 0.38-to-1 (72.25%)
21 Outs 1.2-to-1 (44.68%) 1.2-to-1 (45.65%) 0.43-to-1 (69.94%)
20 Outs 1.3-to-1 (42.55%) 1.3-to-1 (43.48%) 0.48-to-1 (67.53%)
19 Outs 1.5-to-1 (40.43%) 1.4-to-1 (41.30%) 0.54-to-1 (65.03%)
18 Outs 1.6-to-1 (38.30%) 1.6-to-1 (39.13%) 0.60-to-1 (62.44%)
17 Outs 1.8-to-1 (36.17%) 1.7-to-1 (36.96%) 0.67-to-1 (59.76%)
16 Outs 1.9-to-1 (34.04%) 1.9-to-1 (34.78%) 0.75-to-1 (56.98%)
15 Outs 2.1-to-1 (31.91%) 2.1-to-1 (32.61%) 0.85-to-1 (54.12%)
14 Outs 2.4-to-1 (29.79%) 2.3-to-1 (30.43%) 0.95-to-1 (51.16%)
13 Outs 2.6-to-1 (27.66%) 2.5-to-1 (28.26%) 1.1-to-1 (48.10%)
12 Outs 2.9-to-1 (25.53%) 2.8-to-1 (26.09%) 1.2-to-1 (44.96%)
11 Outs 3.3-to-1 (23.40%) 3.2-to-1 (23.91%) 1.4-to-1 (41.72%)
10 Outs 3.7-to-1 (21.28%) 3.6-to-1 (21.74%) 1.6-to-1 (38.39%)
9 Outs 4.2-to-1 (19.15%) 4.1-to-1 (19.57%) 1.9-to-1 (34.97%)
8 Outs 4.9-to-1 (17.02%) 4.8-to-1 (17.39%) 2.2-to-1 (31.45%)
7 Outs 5.7-to-1 (14.89%) 5.6-to-1 (15.22%) 2.6-to-1 (27.84%)
6 Outs 6.8-to-1 (12.77%) 6.7-to-1 (13.04%) 3.2-to-1 (24.14%)
5 Outs 8.4-to-1 (10.64%) 8.2-to-1 (10.87%) 3.9-to-1 (20.35%)
4 Outs 10.8-to-1 (8.51%) 10.5-to-1 (8.70%) 5.1-to-1 (16.47%)
3 Outs 14.7-to-1 (6.38%) 14.3-to-1 (6.52%) 7.0-to-1 (12.49)%
2 Outs 22.5-to-1 (4.26%) 22.0-to-1 (4.35%) 10.9-to-1 (8.42%)
1 Out 45.9-to-1 (2.13%) 45.1-to-1 (2.17%) 22.3-to-1 (4.26%)

[JJ] “I can’t memorize all those numbers!”, Joey the Juvenile complained.

[BB] “I knew you’d say that. Now you know why should learn a formula to calculate the percentage. There are several formulas you can use for just the turn, just the river, and combined, so I’ll let you pick whichever ones you’re comfortable with:”

Accuracy Turn River Turn+River Comments
Excellent Outs times 2.13% Outs times 2.17% Turn odds plus River odds minus Turn odds times River odds2 More accurate than you’ll ever need
Very Good Outs times 2-1/8% Outs times 2-1/6% Outs times 4%, minus (Outs – 8) if 9 or more Outs3 As accurate as you’ll ever need
Good Outs times 2%, plus 1% if 5+ Outs, plus 2% if 13+ Outs Same as Turn Outs times 4%, minus (Outs – 8) if 9 or more Outs3 The easiest reasonable approximation
Bad Outs times 2% plus 1% Same as Turn Outs times 4% Not good for low or high numbers of outs

[JJ] “Which formulas do you use?”

[BB] “I like the ‘Very Good’ approximations. Say I have 13 outs. It’s easy enough to say that’s 13 * 2 + 2 = 28% on the turn (actually three-eighths of a percent less, but I always round), the same on the river (28-1/6% if you don’t round), and 13 * 4 – (13 – 8) = 47% combined. The turn and river numbers are very close, while the combined number is off by only a bit over a percent, so they’re more than good enough.”

[BB] “One additional note, you can count a backdoor flush or a backdoor straight as 1 out for the turn and river combined, since they’re both around 4%.”

[BB] “Once you know the odds, you can compare them to the pot odds or implied odds to see if calling makes sense, which is an explanation for another day…”

Footnotes:

  1. The suits, being unknown, are ignored. Yes, one of them could be a heart, so you really have only 8 flush outs, but all cards that you can’t see are treated equally (any card that’s in another player’s hand or in the muck could still be in the deck).
  2. For example, using the Good approximation, if you have 9 Outs, then you have a 19% chance on the turn, a 19% chance on the river, and 38% minus 19% of 19% (close to 20% of 20%, or 4%) equals a 34% chance on the turn and river combined.
  3. From David Solomon, mentioned in Harrington on Hold ’em: Volume 2.
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