Pot odds

The pot odds (engl. For pot-odds ) are of poker players calculations used to indicate whether the number of inserts is statistically viable. They are usually given in percentages or ratios and are part of a poker strategy . While the pot odds only describe a relationship between the stake and the possible win, the term odds used in this context describes a real probability value . The odds describe the probability of improving your previous hand , which can be estimated with the outs . The outs denote the number of cards that improve your hand. By comparing the odds with the pot odds , one can determine to what extent paying the stake is profitable.

It should be noted that the estimates in this article are only valid on average for a sufficiently high number of games due to the empirical law of large numbers . If you only look at a single game, you cannot make any statements due to the random factor.

Pot odds of the poker variant Texas Hold'em

Calculation of outs, probability of winning and odds

As outs refers to the number of to improve the current hand -capable cards to get a winning hand enabled.

For example, if you have A ♥ K ♥ on hand and there are 3 ♠ 5 ♥ 7 ♥ on the flop , you need another heart card to turn the flush draw into a full flush . There are 13 hearts in the entire game. Four of them (two on hand, two on the board ) are already there. The remaining nine hearts are now the outs .

When Odds is defined as the probability of one of the missing, out to get cards.

Since you know your hole cards and the flop, there are 47 cards left after the flop of 52, which contain the outs.

The odds of improving your cards with the turn card are:

${\displaystyle P\;({\text{Verbesserung durch den Turn}})={\frac {\text{Outs}}{47}}\approx 2{,}13\cdot {\frac {\text{Outs}}{100}}}$

If the turn is on the table, there are still 46 unknown cards. So the probability of improving your cards with the river card is almost the same:

${\displaystyle P\;({\text{Verbesserung durch den River }})={\frac {\text{Outs}}{46}}\approx 2{,}17\cdot {\frac {\text{Outs}}{100}}}$

From the number of outs , you can use the so-called rule of thumb to determine the percentage probability of getting these outs:

${\displaystyle {\begin{alignedat}{2}{\text{Prozentsatz Verbesserung durch eine Karte}}&\approx 2\cdot {\text{Outs}}+1\\{\text{Prozentsatz Verbesserung durch zwei Karten (Turn und River)}}&\approx 4\cdot {\text{Outs}}\end{alignedat}}}$

Since probabilities with around 8 outs are particularly interesting, the first formula is a good approximation for both the turn and the river card. The probabilities for an improvement by the turn or river card will be derived later. The following table gives an overview of the rules of thumb for the particularly interesting hands in poker:

Important chances of improvement after the flop / turn

Current hand	Outs	Probability turn + river rule of thumb	Probability turn + river mathematically exact	Probability river rule of thumb	Probability river mathematically exact
Flush draw z. B. A ♥ K ♥ flop: 3 ♠ 5 ♥ 7 ♥	9 2 ♥ 3 ♥ 4 ♥ 6 ♥ 8 ♥ 9 ♥ 10 ♥ B ♥ D ♥	${\displaystyle (4\cdot 9)\,\%=36\,\%}$	34.97%	${\displaystyle (2\cdot 9+1)\%=19\,\%}$	19.57%
Open Ended Straight Draw e.g. B. 10 ♥ B ♣ Flop: 8 ♠ 9 ♦ 2 ♥	8 7 ♦ 7 ♥ 7 ♠ 7 ♣ D ♦ D ♥ D ♠ D ♣	${\displaystyle (4\cdot 8)\,\%=32\,\%}$	31.45%	${\displaystyle (2\cdot 8+1)\,\%=17\,\%}$	17.39%
Double gutshot z. B. 10 ♥ D ♣ Flop: 6 ♠ 8 ♦ 9 ♥	8 7 ♦ 7 ♥ 7 ♠ 7 ♣ B ♦ B ♥ B ♠ B ♣	${\displaystyle (4\cdot 8)\,\%=32\,\%}$	31.45%	${\displaystyle (2\cdot 8+1)\,\%=17\,\%}$	17.39%
Gutshot z. B. 10 ♥ D ♣ Flop: 5 ♠ 8 ♦ 9 ♥	4 B ♦ B ♥ B ♠ B ♣	${\displaystyle (4\cdot 4)\,\%=16\,\%}$	16.47%	${\displaystyle (2\cdot 4+1)\,\%=9\,\%}$	8.70%
Flush Draw + Open Ended Straight Draw e.g. B. 10 ♥ B ♥ flop: 8 ♠ 9 ♥ 4 ♥	15 = 9 + 8 - 2 2 ♥ 3 ♥ 5 ♥ 6 ♥ 7 ♥ 8 ♥ D ♥ K ♥ A ♥ 7 ♦ 7 ♠ 7 ♣ D ♦ D ♠ D ♣	${\displaystyle (4\cdot 15)\,\%=60\,\%}$	54.12%	${\displaystyle (2\cdot 15+1)\,\%=31\,\%}$	32.61%

With combined cards like a flush draw and an open ended straight draw , you can only count the cards that improve both draws. In our example, the 7 ♥ and the D ♥ improve both the flush and the open ended straight draw. So you don't have 17 outs that you get by adding the nine from the flush draw plus eight from the open ended straight draw. With many outs, the rule of thumb for the turn or river (one card) results in a percentage that is too high (small).

The probability of an improvement through the turn or river card is obtained from the counter-event . So you calculate the probability that none of the outs (we denote them here with O) are visible on the turn as well as on the river :

${\displaystyle {\begin{aligned}P({\text{Verbesserung nach dem Flop}})&=1-P({\text{keine Verbesserung nach dem Flop}})\\&=1-\left({\frac {47-O}{47}}\cdot {\frac {46-O}{46}}\right)\end{aligned}}}$

47 cards are still unknown after the flop, cards do not improve the hand. After the turn only 46 cards are unknown, of which cards do not improve the hand. So the probability of improving your cards on the turn or river is: ${\displaystyle 47-O}$ ${\displaystyle 46-O}$

${\displaystyle {\begin{aligned}P({\text{Verbesserung nach dem Flop}})&=1-{\frac {47\cdot 46-(46+47)\cdot O+O^{2}}{47\cdot 46}}\\&={\frac {(46+47)\cdot O-O^{2}}{47\cdot 46}}\\&=4\cdot O{\frac {1}{100}}-{\frac {O\cdot (O-6{,}52)}{47\cdot 46}}\\&\approx 4\cdot O{\frac {1}{100}}\end{aligned}}}$

After you have determined the probability of winning against a hand suspected by your opponent such as Top Pair Top Kicker, you have to bet this against the stake to be paid relative to the profit to be achieved in order to determine whether the stake is worthwhile.

Example for the betting behavior with an increasing number of players:

Be the pot for the flop, the bet to be called and the number of players (including yourself) who have already paid. You hold Aces ♥ and 9 ♥ and the flop is D ♥ , 2 ♥ , 8 ♠ . The probability of completing the draw is around . We want to know up to what amount you can win at least in the long term . The function of the ratio (profit) / (bet) depending on the investment is: ${\displaystyle P}$ ${\displaystyle x}$ ${\displaystyle n}$ ${\displaystyle x}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle x}$

${\displaystyle {\begin{aligned}f_{n}(x)={\frac {P+nx}{3x}}={\frac {P}{3x}}+{\frac {n}{3}}\end{aligned}}}$

Since strictly decreases monotonically, so all bets with suited to long term achieve a positive profit. For is . So don't call more than the pot. Amazingly, the maximum bet that can be called is already infinite. That means: if someone has bid any amount and this amount has already been called, you should definitely call or reraise. For now, the situation is only improving. Caution: Of course, this consideration does not take into account the fact that if the draw is not completed in the turn, bids will be made again and whether such calls make sense in Sit and Go from a certain blind height. ${\displaystyle f_{n}}$ ${\displaystyle x\leq x_{0}}$ ${\displaystyle f_{n}(x_{0})=1}$ ${\displaystyle n=2}$ ${\displaystyle x_{0}=P}$ ${\displaystyle n=3}$ ${\displaystyle x_{0}=\infty }$ ${\displaystyle x}$ ${\displaystyle n>3}$

In general, the following applies to calculated probability : ${\displaystyle \sigma }$

${\displaystyle {\begin{aligned}f_{n}(x)=\sigma {\frac {P+nx}{x}}=\sigma {\frac {P}{x}}+\sigma n\end{aligned}}}$

If you want to calculate the maximum amount to be called, you bet and then move. ${\displaystyle x_{0}}$ ${\displaystyle f_{n}=1}$ ${\displaystyle x}$

${\displaystyle {\begin{aligned}x_{0}={\frac {\sigma P}{1-\sigma n}}\end{aligned}}}$

If it is less than zero or infinite, you can always call. ${\displaystyle x_{0}}$

Odds notation of the probability of winning

It is just a different way of writing the probabilities introduced above:

Odds = unknown cards without improvement: helpful cards.

An open ended straight draw (after the turn) has the following odds:, because eight of the 46 unknown cards are helpful. The advantage of this notation is that it is easier to determine whether calling makes sense. If the pot in relation to the stake that is to be brought is larger than the odds shown, then the hand is playable. In the first example, there is € 5 in the pot and you would have to bring € 1 to play profitably. So you have pot odds of 5: 1, which corresponds to the odds of 5: 1. Decision-making is easier to apply in practice (if you have memorized the odds in this way). Here is an overview of a few interesting hands: ${\displaystyle (46-8):8=38:8\approx 5:1}$

Current hand	Outs	Odds turn + river	Odds a card
Flush Draw and Open Ended Straight Draw	15th	0.9: 1	2.1: 1
Flush Draw and Gutshot	12	1.2: 1	2.8: 1
Flush draw	9	1.9: 1	4.1: 1
Open ended straight draw or double gutshot	8th	2.2: 1	4.8: 1
Gutshot	4th	5.1: 1	10.5: 1
Expand a pair in the starting cards to three of a kind	2	11: 1	22.5: 1

Pot odds

When pot odds is defined as the ratio between the amount necessary to pay a bet and the current value of the pot . In contrast to the odds , the pot odds are not probabilities, but only the ratio between the bet to be made and the possible profit. The lower the value of the pot odds , i.e. the less money you have to bet to win a certain amount, the better.

If an opponent bets € 1 in a € 5 pot after the flop , the current value of the pot is € 6. You would have to pay € 1 yourself to stay in the game. The pot odds are therefore here 1: 6. So our stake would be one seventh of the resulting pot, or expressed as a percentage, 14.29%.

Odds and pot odds on the call

If you compare the odds with the pot odds, you can make it easier for yourself to bet ( call ) or fold ( fold ). If the probability of improving your hand is greater than the relative proportion of the total pot, you should bet. In the long run you will be on the winning side.

Small rule of thumb:

• If the odds are greater than or equal to the pot odds, you should bet.
• If the odds are smaller than the pot odds, it is better to fold.

Example:

You have an open ended straight draw after the turn with odds of 17% according to the rule of thumb (or 17.39% real). The pot is € 4. Someone bids € 1, which is a quarter of the pot. Does it make sense to call this bet, assuming that the opponent has one or more pair (s) or three of a kind and we are the last player to bet / call? With a stake of 1 € you get the chance to win 6 €. So the bet is 1 / (4 + 1 + 1) = 1/6 = 16.67% of the resulting pot. So the odds are bigger than the pot odds and it makes sense to call this bet. In 100 games in this situation you would bet 100 times € 1, i.e. € 100, and in 17.39% of the cases you would win a pot of € 6, i.e. € 104.34. The expected profit is positive at € 4.34.

The odds notation makes it easier to determine whether a call is profitable. The odds of 4.8: 1 indicate a slightly higher probability 1 / (4.8 +1) = 17.2% than the pot odds of 5: 1 (1 / (5 + 1) = 16.67%).

If the pot were only € 3 (and someone also bids € 1), a call would not be profitable because the stake in the total pot increases to 1 / (3 + 1 + 1) = 1/5 = 20%. The pot odds are higher than the 17% odds. In 100 games in this situation you would bet 100 times € 1, i.e. € 100, and in 17.39% of the cases this time you would only win a pot of € 5, i.e. a total of only € 86.95. So we lose a total of € 13.05 with a hundred games (and a total stake of € 100). The pot odds are now 4: 1, which is a larger ratio than the odds of 4.8: 1.

If further betting rounds can be excluded, the odds after the turn and river are often used. After the flop you have a flush draw with odds of 36% according to the rule of thumb (or 34.97% real). The pot is € 1. An opponent places a bet equal to the pot. Does it make sense to call this bet if you have to go all in? In the event of a win, the stack would triple. A playable situation with a 36% chance of winning. Again, the odds notation makes the decision easier for us. The odds with 1.9: 1 denote (with 34.5%) a higher probability than the pot odds with 2: 1 (33.3%).

Betting odds and pot odds

In the above calculations, for the sake of simplicity, it is assumed that you are sitting behind an opponent and that he has a winning hand. In general, one can win in the following ways:

1. The opponent or opponents fold
2. You have a better hand than your opponents (and you won't get beaten in the process)
3. You complete your hand (and are not hit in the process).

The odds only take into account the probabilities in the latter case when we complete our hand. So there are additional opportunities to win, and mathematically speaking, it makes sense to place higher stakes. The amount to be set (brought) must be the

1. Number of opponents
2. Probability that the opponent (s) will fold
3. Probability that we already have a better hand than the opponents

consider. If there is only one opponent, it is often profitable (especially after the flop) to choose a bet (bet or call) that is twice or even two and a half times as large as the actual odds. This means a bet of 3/4 or half the pot on a draw. With the increased stake it is hoped to increase the likelihood of folding your opponent. The increased probability of folding the opponent increases the probability of an immediate win. Even if you already have the better hand, you increase the expected profit.

Implied odds

This calculation does not include the current pot, but estimates what the final pot will be. The difference arises from the expected stakes of the other players in the following betting rounds.

Implied odds therefore always contain a speculative element, namely in the question: How much bigger will the pot be if I complete my draw at the end?

S      - The maximum amount payable.
C      - Percentage probability of improving the hand.
P      - Estimated final pot size

${\displaystyle S=C\cdot P}$

Moves in no-limit games, for example, are often justified with given implied odds, since in the best case scenario one can assume that the opponent's entire stack will be won in the further course of the game. For this reason, you usually bet or call a draw higher than the odds would suggest (e.g. 3/4 pot size).

Reverse implied odds

Reverse implied odds are the probability of not holding the winning hand even though you get one of the cards that you count towards your outs. In these situations the outs are devalued or reduced. In the example of the open ended straight draw in the table. Hand: 10 ♥ B ♣ Flop: 8 ♠ 9 ♥ 2 ♥ Here you can not fully count the outs 7 ♥ D ♥ because of a threatened flush. This danger must be taken into account when considering whether a call is profitable. To do this, the outs are reduced depending on the hand held, the flop and the number of opponents. If higher flushes are possible with a flush draw, you reduce your outs accordingly. Outs that (with the opponent) lead to a higher road cannot be fully rated either. The so-called texture of the leaf must also be taken into account. If you have a straight draw, you should also reduce your outs on a flop with at least two cards of the same suit. If many opponents join the hand, the probability increases that at least one opponent has a better hand.

Protection of a hand

This is a bet on a hand that is strong but can be beaten later in the game, such as a threatened flush or straight. You should bet so much that your opponents have to place a bet for which they do not have the necessary odds. If you hold (on the turn) the highest pair with the highest card (top pair top kicker), but two cards of the same suit are on the table and you place a bet equal to a quarter of the pot, someone who wins matching flush draw has a playable situation. He gets pot odds of 5: 1, but a flush draw has odds of 4.1: 1. He only needs to bet one fifth of the pot, but wins the pot in (slightly less than) one in four cases. Against an opponent with an impending flush you should bet more than the 3.1th part of the pot (i.e. the probability in odds notation reduced by one), then the call is no longer profitable for the opponent. If the opponent calls every time, he will lose money in the long run. When protecting a hand, it is important to evaluate the possible pot odds of the opponents.

Individual evidence

1. ↑ poker-institut.org Texas Hold'em Outs and Odds ( Memento of the original from January 29, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 89 kB) 
2. ↑ Implied Odds In: Glossary from PokerStrategy.com .