Module 6 · Lesson 29 of 39 · Free
Counting outs: clean and tainted
Nine outs with two cards to come hit 35.0% of the time, and the quick rule of 4, outs × 4%, says 36%: close enough to decide at the table. But a card is an out only if it makes you the winner. A heart that gives you a flush and gives him a full house is not an out.
The hand
The turn. You hold A♥ J♥ with K♥ 9♥ on the board: a flush draw. You are sure the Tight-aggressive player has a set of nines, 9♣ 9♦.
Predict first
How many outs do you have for the river?
- 9: every heart that is left
- 7: the hearts that do not pair the board
- 12: the hearts and the three aces
Show the answer
Answer: 7: the hearts that do not pair the board
Nine hearts are unseen, but the 4♥ and the 2♥ pair the board and give his set a full house, nines full, which beats your flush: 9 − 2 = 7 clean outs. An ace gives you a pair of aces, still behind three nines.
Working it out
With one card to come, divide the outs by the cards you have not seen: 52 − 2 − 4 = 46 on the turn, his two among them. Your 7 clean outs hit 7 ÷ 46 = 15.2% of the time; all nine hearts would have said 9 ÷ 46 = 19.6%. With two cards to come, count the misses, as in the lesson on the complement rule. From the flop, 9 outs miss the turn 38 times in 47 and then the river 37 times in 46, so they miss both (38 ÷ 47) × (37 ÷ 46) = 65.0% of the time and hit 100% − 65.0% = 35.0%. The rule of 2 and 4 is the shortcut: outs × 2% for one card, 18% for 9 outs, and outs × 4% for two cards, 36%.
Clean outs win against the hand you put him on; tainted outs complete your hand but improve his even more. Count only the clean ones. Then use the chance that matches the cards you will see for the price: outs ÷ 47 on the flop or outs ÷ 46 on the turn for one card, and the two-card chance only when you will see the turn and the river for this one call. The rule of 4 is within a point of the exact chance up to 8 outs and about a point high at 9; above that it runs higher, 60% for 15 outs against an exact 54.1%.
Formula
One card: outs ÷ 47 on the flop, outs ÷ 46 on the turn. Two cards: 1 − (47 − outs) ÷ 47 × (46 − outs) ÷ 46. Quick rules: outs × 2% or outs × 4%
Check your understanding
On the turn you have 6 clean outs. What is your chance to hit on the river, in %?
Show the answer
Answer: 13%
6 ÷ 46 = 13.0%. The rule of 2 says 6 × 2% = 12%.
On the flop you have 12 outs, and you will see both the turn and the river. What is your exact chance to hit, in %?
Show the answer
Answer: 45%
You miss both cards (35 ÷ 47) × (34 ÷ 46) = 55.0% of the time, so you hit 100% − 55.0% = 45.0%. The rule of 4 says 48%.
You have a flush draw and believe he has two pair, made with both of his cards. Which of your flush cards are not clean outs?
- Those that match a rank of his two pair: they give him a full house
- Those higher than your own cards
- None: a flush always beats two pair
Show the answer
Answer: Those that match a rank of his two pair: they give him a full house
A card that matches one of his pairs gives him three of that rank with his other pair: a full house, which beats your flush. Every other card of your suit wins.