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Poker Mathematics: Outs & Odds

The intricacies of card games like poker involve a lot of mathematics, the odds being the primary ones. Knowing the odds in poker is vital in the long run, as these help master the game and adjust your poker strategy. Our experts have created the following guide where they explain the poker odds and outs that you face playing Texas Hold ‘em poker and will also discuss the math side of poker and how to use the odds to make the right decision. But before we elaborate on poker odds, we need to explain poker outs and what these mean!

Poker Outs: Definition

You will often face an incomplete hand in poker that has a good chance of becoming a made hand by drawing one card. The card that, when drawn, makes a made hand is called the out. It is straightforward to count the outs, as you basically need to count the number of unseen cards that can improve your hand. But there are some rules to this, especially in Texas Hold ‘em poker.

The game starts with a single 52-card deck, and you get two hole cards, and the flop reveals three community cards on the table, leaving you with 47 unseen cards. So, for example, you have a 6 ♣ and an 8 ♣, and the flop reveals a 10 ♣ and 3 ♣, and 9 ♥. The probability of hitting a flush is very high, as you only need one more ♣ card to get a flush. From the total of 13 ♣ cards in the deck, you have two, and the flop shows two, so there are 9 other ♣ cards in the deck, which means you have nine outs to make a hand.

The poker outs calculations are done based on all the unseen cards, as you don’t know what cards your opponents hold. Still, calculating the outs is important for adjusting your Texas Hold ‘em strategy.

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As you can see, you can determine the outs after the flop is dealt. The more outs a draw has, the stronger it is and the higher the probability of hitting your hand. Depending on your hole cards and the flop, the outs vary, and you should never overcalculate the outs. If your hole cards and the flop cards have multiple possibilities, then you need to calculate the actual number of outs. To use the same example as above: you have a 6♣ and 8♣, and the flop reveals a 10♣, a 3♣, and a 9♥. In this case, you only need any ♣ card to hit a flush, where there are 8 outs, but you can also make an open straight if the following streets reveal a 7 and a J. Thus, the actual outs count is not 17 (8 outs + 9 outs), but it is 15, as you need to remove the possible outs of hitting a 7 or a J twice.

Poker Odds: Calculating Poker Odds Based on Outs

The odds in Texas Hold’ em refer to the probability of hitting an out or the ratio of the number of ways not to draw a hand to the number of ways to draw the hand. The odds are relatively straightforward to calculate, and you don’t need to be a math prodigy to do this.

However, you will need to know all the poker hands by heart and all the probable outs that can help improve any hand. The following table shows the number of outs and probabilities for possible draws on the turn and the river:

Possible draw Outs Probability to hit on the turn Odds to hit on the turn Probability to hit on the river Odds to hit on the river Probability to hit on the turn or river Odds to hit on the turn or river
Four of a kind 1 2.1% 46 to 1 2.1 % 45.0 to 1 4.3% 22.5 to 1
Three of a kind 2 4.3% 22.5 to 1 4.3% 22.0 to 1 8.4% 10.9 to 1
High pair 3 6.4% 14.7 to 1 6.5% 14.3 to 1 12.5% 7.01 to 1
Insight straight draw 4 8.5% 10.8 to 1 8.7% 10.5 to 1 16.5% 5.07 to 1
Two-pair or three-of-a-kind 5 10.6% 8.4 to 1 10.8% 8.2 to 1 20.3% 3.91 to 1
Any pair with your holding 6 12.8% 6.83 to 1 13% 6.67 to 1 24.1% 3.14 to 1
Insight straight or high pair 7 14.9% 5.71 to 1 15.2% 5.57 to 1 27.8% 2.59 to 1
Open-ended straight draw 8 17% 4.88 to 1 17.3% 4.75 to 1 31.5% 2.18 to 1
Flush draw 9 19.2% 4.22 to 1 19.5% 4.11 to 1 35% 1.86 to 1
Inside straight or a pair 10 21.3% 3.7 to 1 21.7% 3.60 to 1 38.4% 1.60 to 1
Open-ended straight or high pair 11 23.4% 3.27 to 1 23.9% 3.18 to 1 41.7% 1.40 to 1
Flush or high pair 12 25.5% 2.92 to 1 26% 2.83 to 1 45% 1.22 to 1
13 27.7% 2.62 to 1 28.2% 2.54 to 1 48.1% 1.08 to 1
Open-ended straight or a pair 14 29.8% 2.36 to 1 30.4% 2.29 to 1 51.1% 0.955 to 1
Flush or open-ended straight 15 31.9% 2.13 to 1 32.6% 2.07 to 1 54.1% 0.848 to 1
16 34% 1.94 to 1 34.7% 1.88 to 1 57% 0.755 to 1
17 36.2% 1.76 to 1 37% 1.71 to 1 59.8% 0.673 to 1
Flush, inside straight, or a pair 18 38.3% 1.61 to 1 39.1% 1.56 to 1 62.4% 0.601 to 1
19 40.4% 1.47 to 1 41.3% 1.42 to 1 65% 0.538 to 1
20 42.6% 1.35 to 1 43.3% 1.3 to 1 67.5% 0.481 to 1
Flush, open-ended straight, or pair 21 44.7% 1.24 to 1 45.6% 1.19 to 1 70% 0.430 to 1

The poker odds give the winning probability of hitting an out, and in general, the odds in Texas Hold ‘em result from dividing the outs by the number of possible cards. These are often referred to as “x to y.” For example, if you have nine outs and there are 47 possible cards, the odds are 9/47=0.19 or 19%. It equals 4 to 1, meaning you get four chances to lose against one chance to win.

Here, we also need to mention the pot odds, as you only need to calculate the ratio of the possible reward compared to the stake you need to place or calculate a risk/reward ratio. Please check out the guide where we discuss the poker pot odds in detail. Here, we’ll only mention that the pot odds determine whether it is worth it to call a bet.

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Using Odds to Make a Decision: Bet or Fold

Drawing to improve the poker hand is an integral part of poker, and it is often down to the decision of whether to call a bet based on the odds, which can also help you decide to place a c-bet. We’ll use the same example as above, where we calculated that there are 4 to 1 odds to win the bet, but the decision needs to take into account the pot odds.

Here is an example that we believe will clear things for you:

  • Let’s say you play at a pot with $90
  • our opponent places a $10 bet, putting the pot size to an even $100
  • You need to match the $10 bet to get to the river,
  • Thus, you stand to win $100 on a $10 bet, giving you 10 to 1 winning odds.

Since you have a 4 to 1 chance to draw a flush, the odds are in your favor, as the pot odds are higher than the draw odds.

However, if your opponent raises the bet to $30, then the pot becomes $120, and you’ll need to pay $30 to call their bet. After calculating the pot odds, the resulting pot odds are 120/30=4, giving you an approximate 4 to 1 pot odds. This means that it will not be profitable to call the bet, as the pot odds are equal to the draw odds, and you will lose in the long run.

You stand to win money if the pot odds are higher than the draw odds of the incomplete hand, and if the pot odds are equal or lower than the draw odds of the incomplete hand, you will lose in the long run.

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Why calculate outs in poker?

Calculating the outs in poker is helpful to determine whether your hole hand has potential and whether it will be profitable to call a draw. You can use the outs as a guide and decide whether you should fold, call, or place a bet.

How to calculate poker odds?

You need to divide the number of outs by the number of cards, which will give you the poker odds. In Texas Hold ‘em poker, you need to divide the number of outs for the given hand and the number of unseen cards; thus, the odds will change on the flop, the turn, and the river.

What are pot odds?

The pot odds refer to the ratio of the possible winning amount to the bet you need to place. If the pot odds are higher than the odds of drawing an incomplete hand, then you stand to win in the long run, and if the pot odds are lower, then you stand to lose in the long run.

How do you calculate poker pot odds?

The pot odds are the ratio between the possible reward and the bet you need to place, meaning you need to calculate the amount in the pot and the bet you need to call in order to make the draw. If this ratio is higher than the odds of making an incomplete hand, you should call the bet, and if the ratio is lower, you are better off folding.

What are good odds?

The good odds in poker mean that the pot odds are higher than the draw odds of making your incomplete hand. In other words, if you have better odds of making an incomplete hand than the pot odds, then you will lose money in the long run, but if you have better pot odds compared to the draw odds of making an incomplete hand, you will win money in the long run.