Poker Probabilities: Counting Odds and Rule of Thumb

Poker Probabilities: Counting Odds and Rule of Thumb

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For sure you will already have heard of the term “outs” in relation with playing poker. But what do “Outs” mean? The principle of outs is quite simple. Any card that helps you to shape the winning hand is called an out.

Typically, we speak about outs after the flop. After the flop, it is very easy to exactly calculate the probability to win. And using a simple rule of thumb, it is even easier to calculate the probability to win the hand. Let’s have a look at three examples.

Outs Example 1: Straight Draw on the Turn

Suppose your hole cards are 9 and 8 of hearts. On the board are an ace, a ten, a seven and a three in different colors. A jack or a six would help you to make your straight, which will be very likely the winning hand. How many outs do you have?
The answer is eight. Four jacks and four sixes are among the unknown cards. The probability that one of those eight cards will be shown on the river can be calculated easily. 46 unknown cards remain in the game: from the 52 cards, two are your hole cards and four of them are on the board. Of course, other players also hold cards in their hands, and some cards are in the “muck”. But because you do not know these cards, you have to consider these cards as well for this calculation. So you have 8 outs out of a total of 46 cards. The probability that you will make your straight on the river is therefore 8 / 46 = 17.4%.

Outs Example 2: Two Pairs on the Turn

Suppose you hold ace-king of hearts. On the board are an ace, a king, a seven and a four, all in spades. The likelihood is high that with your top two pairs, you lie behind behind an opponent who already has a flush. The only thing that could help you to win this hand is a full house. How many outs do you have?
The answer is four. Each ace or king of diamonds or clubs will make you a full house. Your winning percentage is thus 4 / 46 = 8.7%.

Outs Example 3: Open-ended Straight and Flush Draw on the Flop

As a final example; let’s have a look at one of the most promising situations to improve your hand. Suppose you hold king and queen of hearts. The flop shows jack and ten in hearts, and a three in spades. Each ace or nine helps you to complete your straight. And every heart will make you a king-high flush, the probable winning hand. How many outs for the turn do you have?
The answer is fifteen. Eight cards help you to make a straight and nine cards will make you a flush. This adds up to 17, you might think now. But you have to consider that you can count each card only once. The eight cards that complete your straight already include the ace and nine of hearts. Thus, only seven more cards will help you to complete your flush. Your chance to shape the winning hand on the turn is therefore 15/47 = 31.9%.
For advanced poker players: what is the probability of making the winning hand from the flop to the river (straight or a flush) in the above situation? This question is important when you have to decide whether to call an all-in on the flop or to fold your hand.
The answer is 54.1%. To calculate the probability, the simplest approach is to calculate the probability of the complementary event first and to multiply them. The chance not to make the winning hand on the turn is 32/47 (32 of 47 cards do not help you). The chance not to make the winning hand on the river is 31/46 (31 of 46 cards do not help you). You just multiply these two probabilities: 32/47 x 31/46 = 45.9%.The chance of not making the best hand on the turn and on the river is therefore 45.9%. So the probability to make the winning hand is 100% – 45.9% = 54.1%. This means that on the flop, having both an open-ended straight and a flush draw, you can call an all-in regardless of pot odds, because you will win in more than 50% of cases.

Table Probabilities (Outs, Flop to River, Turn to River)

Outs River Turn + River Example
1 2.2% 4.4%
2 4.3% 8.4%
3 6.5% 12.5%
4 8.7% 16.5% Gutshot Straight Draw (eg, A-K and J-T)
5 10.9% 20.3%
6 13.0% 24.1% Two Overcards (eg, A-K with 8-5-2 on the board Board)
7 15.2% 27.8%
8 17.4% 31.5% Open Ended Straight Draw (eg, 9-8-7-6)
9 19.6% 35.0% Flush Draw (eg, A-K-7-4 in hearts)
10 21.7% 38.4%
11 23.9% 41.7%
12 26.1% 45.0% Flush plus Gutshot Straight Draw
13 28.3% 48.1%
14 30.4% 51.2%
15 32.6% 54.1% Flush plus Straight Draw

Poker Probabilities: Rule of Thumb

For sure you do not want to memorize the above table. And even when the calculation of probabilities is simple, you might not want to do such calculations while playing live at a poker table. The good news is that there is a proven and very simple rule of thumb to calculate the probabilities in the Texas Hold’em poker game:

Outs x 2 + 1 = probability of improvement with one card

Outs x 4 = probability of improvement with the turn or river card

This formula represents reality more than close enough. If you have an open-ended straight draw (8 outs), then the rule of thumb for the river would give you percentage of winning of 17% (2 x 8 + 1). Calculated exactly, the probability is 17.4%. In an all-in situation on the flop the rule of thumb gives you a chance of winning of 36% (4 x 9), calculated exactly it is 35%.

Conclusion Outs and Odds in Poker

Estimating probabilities in poker is very simple. You just have to count the number of outs (the cards that will help you) and calculate the probability of winning using the above rules of thumb. Any player who is serious about playing poker should at least know those two rules of thumb. If this article sounded difficult to understand, you might want to print it out and read it later again in order to understand the subject of counting odds and calculating probabilities in Texas Hold’em poker even better.