Odds, Outs and Probabilities

Poker is a game based on decisions. The idea is to make as many correct decisions and as few mistakes as possible. The player who makes fewer mistakes than his opponent will win in the long run.

Probabilities, i.e. odds and outs, are the basis for making right decisions and avoiding mistakes. They are extremely helpful when it comes to answering the following questions:

  • What is the probability that I have the best hand?
  • What is the probability that I will get the best hand with one of the upcoming community cards?
  • Do I currently have the better or worse hand?

A player who doesn't base his decisions primarily on mathematical probabilities will not be able to win in the long run.

You could also play hundreds of thousands of poker hands to learn about the occurrence of certain events in a "practical" manner. But the number of hands played has to be practically infinite to ensure that you achieve the right probabilities.

There are however a few mathematical principles that help you to determine precise probabilities. Don't worry, we'll try to keep it as simple as possible.

Let's start with the odds pre-flop: They help you to determine whether you have a good or a bad hand. Especially in the late stages of a tournament where there are often a lot of all-ins before the flop, it is very important to know the odds of beating your opponents' potential hands.

We all know that the best starting hand is AA. Two Aces beat random opponent cards in more than 80% of cases. But unfortunately, we aren't always dealt aces. For this reason we have selected a few typical pre-flop situations.

 

It is recommended that you learn these few values by heart. It will take about half an hour, but it will soon start paying off.

Even more often we will be in a situation where we see a flop with one or more opponents, and then an opponent makes a bet so we have to decide whether or not to call (a raise depends on too many other factors and should be ignored at this point). Since the flop provides far more information than we had before, we can now fairly accurately calculate our odds.

To do this, we need to know the outs, odds and probabilities. Let's start with the outs:

Outs

Outs are all cards in the deck that improve our hand in such a way that we will probably win a showdown. Consequently, outs only have significance, if there is still at least one community card to come. Thus, there are no outs after the river.

  • Let's look at a few examples:

We are holding . The flop is: . So if another heart appears on the turn or river, we will have a flush. Unless another player has a full house or better we will win the hand. The board isn't paired, so none of the opponents could have a full house yet.

 

There are precisely 13 cards of each suit in the deck. We have two of them in our hand, and another two are on the board. Four of the 13 hearts cards have therefore already been dealt, meaning that there are still nine hearts left in the deck. In this case these nine cards are our outs.

  • Here's another example:

We have and the flop is . Now any ace or nine will complete our straight. There are four aces and four nines in the deck, so we have a total of eight outs.

If one card is missing to complete a straight, we have four outs (e.g. hole cards: , flop: , outs: ).

  • Let's look at the next example:

I have , and the board is . One of the four queens in the deck will make my straight. If my opponent has a small pocket pair in his hand, e.g. , then we would have additional outs as now any king or any jack would give us a higher pair. In this case the number of our outs would increase to ten (the 4 queens, 3 kings, and 3 jacks).

If I hit two pair (e.g. with on a board of , there are still four cards that could make me a full house (consisting of three of a kind plus a pair): and .

If I hit a set because I have e.g. , the board is and I'm worried that my opponent has a flush, then there are seven cards that could help me to make a full house or better after the flop (i.e. a seven, one of the three remaining twos and one of the three remaining jacks). If I don't hit any of my outs on the turn, because it brings e.g. the , then I have three additional outs with the three remaining queens and thus ten outs on the river.

  • Here's another example:

I am holding and the board is . Now I have a draw (thus my hand is incomplete and yet worthless, but could become a strong, profitable hand if one or several fitting community cards appear), namely an open-ended straight draw and at the same time a flush draw. This means that I have nine outs to make a flush and eight outs to make a straight. At the same time we have to consider that we counted two cards twice (in this case the and the ), which have to be discounted. In total we only have 9 + 6 rather than 9 + 8 and therefore 15 outs.

Hidden outs

Let's look at the hidden outs, which are hard to discover, since they won't directly improve our hand, but reduce the value of our opponents' hand.

  • Example:

The opponent has , and we have . The board is . Not only will one of the two remaining aces help us, but also one of the three kings or one of the three tens. So we have eight outs and six of them are hidden outs. Why is that? The board would contain a pair if another king or another ten appears, meaning that we now would also have two pair with our pocket aces (the hole cards are also called pocket cards and two aces as hole cards are therefore called pocket aces) together with the pair on the board we now would also have two pair, which would be higher than the two pair of our opponent.

  • Here's another example:

We're holding , our opponent has . The board is . Not only would the three kings and the three aces help us to get a higher two pair, any six or five would help us as well. This is because with a five or six, the board already contains two pair that are both higher than the opponent's pocket threes, meaning that the fifth card, the kicker, would decide the outcome of the hand. Our ace is the best-possible kicker, so we have 12 outs and six of them are hidden.

But what happens, if we and our opponent both have a draw? How will this influence our outs?

Discounted outs

Advanced players don't simply calculate the theoretical outs - they ask themselves what hand the opponent really has and whether one of the cards we hope to appear would give another player, who also is on a draw, an even better hand. 

  • Let's look at the straight draw example again:

We have and the flop is . We have calculated 8 outs so far (four aces and four nines).

How will our outs change if one of our opponents has two hearts in his hand, e.g. and is therefore hoping for a flush? Then two of our outs, i.e. or the , would give our opponent a better hand even if we were to hit our straight. In this case we have to discount both cards from the number of our outs. Consequently, we would only have six outs, which would significantly reduce the probability to win the hand.

In general we take a somewhat pessimistic approach when it comes to discounting outs, i.e. it's better to discount one out more than one too few!

 

Probabilities

Let's take a look at probability calculations and try to work out the probability of hitting one of our outs on the turn or river. This probability is a vital criterion when it comes to making a decision.

Here's a small piece of advice before we start: There are very simple rules of thumb with the help of which we can determine probabilities accurately enough to save all of the cumbersome mathematics. Since it could be reasonable for your own calculations and in order to answer a number of interesting questions you have to be able to precisely calculate the probabilities. First of all we're going to look at the mathematical calculation method now.

Let's return to the example with the flush draw.

We have . The flop is .

A card deck consists of 52 cards. I know both of my hole cards and the three open community cards, known as the flop. So this means:

52 (all cards)

2 (my hole cards)

3 (the flop cards)

= 47 unknown cards.

The number of the players at the table is completely irrelevant here. As long as the opponent doesn't give away which cards he's holding, all of the unknown cards are to be considered a mathematical quantity. They include all of my opponent's cards, cards held by the dealer, and any burn cards (cards that are withdrawn by the dealer and are therefore no longer part of the game) lying covered on the table.

We know that there are exactly 13 cards of each suit in the deck, from two through to the ace. Since I already have two hearts and there are two hearts in the middle, there are another nine hearts cards left in the deck, my outs.

To calculate the probability of the next card being a heart, we have to create a ratio of the nine hearts cards to all of the available cards (47). And this is performed by means of division.

9/47 = 0.191.

Now we have our result. If we want to express this as a percentage, then we have to multiply our result by 100 (simply move the decimal two places to the right), giving us 19.1%. The probability of making a flush on the turn in the above situation is therefore 19.1%.

If no heart appears on the turn, what is the probability of a heart appearing on the river? There are still nine hearts cards in the game, but 46 cards available in total (as the turn now has to be deducted). So this time we divide 9 by 46, giving us 19.6%. The probability has slightly increased, since there is one less "blank" in the game.

We therefore know how to calculate the appearance of one of our outs on the turn or river.

But if I want to know the probability of a hearts card appearing "EITHER on the turn OR/AND on the river" (which is very important in all-in situations), I need to adapt the calculation somewhat. You can't just add up both results. OR-Probabilities can be calculated directly but it's complicated, whereas AND-Probabilities can simply be multiplied. We therefore make an AND-Probability from the OR-Probability, which allows us to calculate the counter-result. So the question now has to be changed to:  What is the probability of "NO heart appearing neither on the turn nor on the river?" This probability is then deducted from 100% and gives us the value we are looking for.

The probability of NO heart appearing on the turn is 38 to 47, as 38 of the 47 cards (47 minus 9) will not be a heart.

The probability that NO heart will appear on the river is 37 to 46 (as we have one heart card and one card less in general).

The probability of no heart appearing on the river and on the turn is simply the product of these two probabilities: 38/47*37/46 = 0.65 = 65%. If I want to use this information to derive the probability of at least one heart appearing on the turn or river, then I simply deduct 65% from 100% = 35%. So 35% is our result.

We can summarise this entire process in a formula:
  • If outs is the number of cards that improve our hand and p(Outs) is the probability of these cards appearing on the turn and/or river, then
  
OutsFormel
 

The rule of thumb 

As only a small number of poker players can or want to calculate this formula off the top of their heads, we will now disclose a secret concerning the rule of thumb mentioned previously:

  • The rule of thumb for the probability of a draw arriving on the turn is:
OutsFormelx2
  • The rule of thumb for the probability of a draw arriving on the turn and/or river is
OutsFormelx4
 

The following applies to our flush draw: The probability of a hearts card appearing on the turn is 9 x 2 + 1 = 19%. The probability of a heart appearing on the turn and/or river is 9 x 4 - 1 = 35%. The actual values would be 19.1% and 35%, so our rule of thumb provides us with a fairly accurate result.

If you have a lot of outs (14 or more), the second rule of thumb doesn't work out very well. In this case you should keep the following in mind: From 14 outs on you are a favourite (51,2%) and should never fold.

Odds

A better way of expressing this in a number of other poker related situations is to describe the probabilities in terms of the so-called odds, which you can also call "profit margin".

  • Odds are the ratio of my non-outs to my outs (i.e. the ratio of possible misses to hits).

In the previous flush draw example (we had and the board was ) we calculated nine outs, which would help us to make the best hand in case they appear. All of the other cards in the deck are of no use at the moment, i.e. they would not assure that we have the best hand. After the flop there are 38 cards (47-9) that would not help us. We therefore have 38 misses and nine hits which can appear on the turn. Our odds are therefore 38 to nine (also written as 38:9). In order to better calculate this, odds can also be shortened, so in our case we would divide both sides by nine, giving us 4.2 to 1. The idea behind this is to always try and get a one after the colon.

Odds therefore describe how often a miss appears compared to a hit. In the long run you will miss your flush on the turn four times as often than you hit it.

To become a successful poker player, you need to be able to quickly calculate odds and outs. The best way to learn this is to play with friends or at play money tables until you can quickly calculate your outs and odds. You will pick up the most important odds fairly quickly. So a flush draw has nine outs and odds of approx. 2 to 1, an open-ended straight draw has eight outs and odds of approx. 2.3 to 1 up to the the river etc. It is again highly recommended that you learn these values by heart.

The table below shows the respective probabilities for one to 20 outs on the flop up to the turn as well as from the flop up to the river.

 

Pot odds

Having extensive knowledge of outs, odds and probabilities is of no use whatsoever if they cannot be compared with the maximum amount of money that I'm allowed to pay in order to be able to make a profit in the long term. Only if these figures are compared with the odds we can make mathematically correct and therefore profitable decisions. This is the pre-requisite for playing profitable poker in the long run. So let's take a look at the so-called pot odds.

  • Pot odds are the ratio of the potential win, i.e. the pot, to the amount of the bet i have to call at a respective point in time. Thus, how much can I win at what prize.

The pot odds are fairly easy to calculate.

  • Example:

Let's say there is $5 in the pot and  my opponent bets $1. When i want to stay in the hand, I also have to call $1 in order to win $6, i.e. the previous pot of $5 plus my opponent's $1 bet. My pot odds are therefore six (which I can win) to one (which I have to call), written as 6:1.

  • A second example:

Let's assume we're playing at a no-limit hold'em table and there is $5 in the pot, as was the case in the first example. My opponent now bets $5, i.e. the amount currently in the pot. I therefore have to call $5 in order to win $10 (the $5 in the pot and the opponent's $5). In this case my pot odds are ten to five, or, when shortened, two to one.

The pot odds now have to be compared to the odds. If the pot odds are higher than the odds you can call profitably.

 

PotOddsFormel

 

  • Example:

Let's assume that there are $30 in the pot, our opponent bets $6 and we have a flush draw on the turn. Thus, we are getting pot odds of 6 to 1 (so we have to risk $6 in order to win $36). However, the odds for making our flush on the turn are approx. 4 to 1. The pot odds are higher and therefore a call is profitable.

If the odds and pot odds are exactly the same, it makes no difference whether to call or fold. In the long run we will not win anything, but we won't lose anything either. It can, however, happen that despite making the right decision, my opponent hits exactly the card he needs to win the hand. This may even happen several times in a row, which is improbable, but not impossible. The main thing here is not to become insecure due to such "bad beats". At the end of the day, your opponent has made a mistake which will be to our advantage if we keep making the right decisions. A beginner can therefore win a hand or even a small session against a star like Daniel Negreanu. However, in the long run the player who makes fewer mistakes and mathematically correct decisions will come out on top.

Implied pot odds

The principles already mentioned describe decisions based on the pot odds at the time in question. So I calculate the odds and compare them to the current pot odds. Advanced players can go one step further: They include into their decision the eventuality that an opponent may make additional bets when they make their hand. Then these "future" bets are added to the pot odds, which can turn a fold into a call. These additional pot odds are known as "implied odds".

  • Implied odds describe the ratio between the bet that has to be called and the potential profit, i.e. the pot, including all future bets as well.

For beginners this would over-complicate the decision-making process, so we will not go into more detail for the moment. Calculating odds and outs provides a player with a solid basis for playing the low limits. If you want to deal with implied odds you can read the following article.

Overview of the different steps

  1. Determining the discounted outs
    After calculating the outs you have to discount the cards, which improve our hand, but potentially give our opponent an even better hand.
  2. Calculating the pot odds
    We have to calculate the ratio of the current pot to the bet we have to call. In order to do this, odds can be shortened. The idea behind this is to get a one after the colon.
  3. Comparison of odds and pot odds
    Check if the odds are higher or lower than the probability to lose (by looking at the odds).
  4. Making a decision
    Depending on the situation you decide whether to call or fold.

 

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