Mathematics is the foundation on which a winning strategy rests. And poker odds are one of the key elements in calculating the profitability of a hand.
This article explains what pot odds are in poker and teaches you how to count the probability of winning in order to make profitable decisions and win more often.
What is poker math for?
Knowledge of math is very important in poker because the strategy of the game is based on odds, probabilities and calculations. In a short period of time, a positive result is possible without a mathematical component. However, at a distance, ignoring this knowledge leads to a loss.
Understanding the math in poker helps:
- Analyze the profitability of a particular action in a given situation. For example, is it worth betting on a poker hand that has not yet been completed, is it profitable to call, or is it better to fold.
- Calculate the probability of a desired card or winning combination on the following streets.
- Calculate the possible win or loss.
- Identify the nuts. Being ahead, you can play as aggressively as possible.
- Minimize bad decision making.
- Win money from weak players who rely only on luck.
What are poker odds
Odds or odds are the odds of the hand strengthening. They are written as the ratio of the number of losing options per winning one.
So the ratio of 5: 1 means that for 5 losses, according to the theory of probability, there will be 1 winning outcome. That is, the poker player will win in 1 hand out of 6. A 1: 5 or 1 to 5 record can also be used.
Based on this ratio, you can determine the profitability of the solution. The procedure is as follows:
- Determine the potential winning combination you can count on.
- Calculate the number of outs for the best combination.
- Discount outs.
- Define your own odds.
- Calculate the pot odds considering the potential actions of the next players.
- Compare the odds for improving the hand and the pot odds. When the odds are higher, you can place or call, otherwise you can fold.
These actions seem complicated only at first glance. Let’s take a closer look at them.
To determine the likelihood of improvement, a player must be able to calculate the number of outs. These are cards that can improve a hand to a winning one.
If a player has A-4 suited and there are 2 more cards of that suit on the table, he has 9 outs to the flush.
In this case, outs are only cards that definitely improve the hand to a winning one. So, if a player catches a second or third pair on a board like A-9-T, he cannot be considered 100% out.
In a general sense, probability is the ratio of the desired outcomes to the total number of outcomes or events. In poker, most often, this indicator reflects the percentage of getting the right out on the turn, river or for two streets at the same time.
Odds are converted to probability using the following formula:
number of winning events / (number of winning events + number of losing situations) * 100%
In fact, these are the same chances, but in percentage terms from 0% to 100%. Some players find it more convenient to present this information as a percentage. So the odds ratio of 1: 3 to the close of the flush means that the probability of this event will be 1/4 or 25%.
The probability of a new event occurring does not depend on the previous result. Therefore, if a straight does not close 5 times in a row, this does not increase the likelihood that the combination will be exactly hit 6 times.
Poker hand odds table
Beginners and players who are poor at poker math can use visual materials during the game – hand tables. They allow you to do without complicated calculations in your head.
Below are several tables with ready-made calculations. Each of them is easy to use while playing.
|Outs||The likelihood of coming out on the turn||The likelihood of coming out on the river||Two streets chance|
The second part has 3 columns:
- Tern. The indicator reflects the ratio for one street and should be used when trading on the flop, if the opponent does not offer to put all-in.
- River. Evaluates the indicator for the river and should be used in trading on the turn.
- Turn and River. The column shows the ratio of losing and winning hands, calculated simultaneously for two streets.
The following table helps you assess the perspectives of finished or semi-finished combinations.
|Outs||Flop to turn||Flop to River||Example|
|one||45: 1||22: 1||Junior set against senior|
|2||22: 1||11: 1||Younger couple versus older|
|3||15: 1||7: 1||One overcard|
|four||11: 1||5: 1||Gutshot|
|five||8: 1||4: 1||Middle pair against high|
|6||7: 1||3: 1||Two overcards|
|7||6: 1||2.5: 1||Gutshot and overcard|
|eight||5: 1||2: 1||Open-ended straight draw|
|nine||4: 1||2: 1||Flush draw|
|10||4: 1||1.6: 1||Gutshot and two overcards|
|12||3: 1||1.2: 1||Gutshot and flush draw|
|fourteen||2.4: 1||1: 1||Straight draw and two overcards|
|fifteen||2: 1||1: 1||Flush draw and two overcards|
|Pocket pair versus pocket pair||KK vs 77||80.5% / 19.5%|
|Pair vs Suited Overcards||66 vs KQs||52.8% / 47.2%|
|Pair Against Offsuit Overcards||66 vs KQo||53.5% / 46.5%|
|Pair Against Low Cards||QQ vs 43o||83.5% / 16.5%|
|Two overcards versus low cards||QK vs 9T||63.9% / 36.1%|
|Low and overcard versus two middle cards||A6 vs 89||55.8% / 44.2%|
|Same cards with different kickers||KQ vs K8||73.3% / 26.7%|
|Pair vs One overcard||JJ vs A6||71.2% / 28.8%|
Poker rooms do not directly prohibit the use of tables, unlike poker calculators, because they are not calculations on the fly. They can be opened in the next window, but it is more convenient to keep them printed. This way, they can be accessed without leaving the poker tables.
How to calculate
Knowing the basic information about all the cards (in the deck there are 4 suits of 13 cards), your hand and the general board, you can make calculations in each draw. The main thing is to correctly determine the number of outs.
For calculations in the form of relations, you can use the formula:
Odds = Inappropriate cards / Cards that will improve the combo
Odds = Cards in the deck other than Outs / Outs
Example: there are 2 participants left in the drawing. One user has A-5 on his hand and KT-8 on the board. The number of outs will be 3, which is 3 aces.
Since 5 cards are already known, 47 will remain unknown.
The odds of the required card on the new street will be: (47–3) / 3 = 14.67: 1.
You don’t have to keep all of the poker odds tables in your head. Simple rules are used for counting:
- Rule of two.
- Rule of four.
If you need to count the odds for improvement from flop to turn or turn to river, you need to multiply the number of your own outs by 2. Since 1 of them gives about 2% improvement: 1/47 * 100% = 2.13%. To calculate the probability on two streets, multiply by 4.
Example: a player has A-7 hearts on the board, T-2 hearts on the board and 3 of a different suit. To improve the combination, you will need 3 aces and any hearts of hearts – there are 9 of them left (13–4).
There are 12 outs in total:
- The odds for improving on the turn according to the rule of two will be: 12 * 2 = 24%;
- The odds to improve on any of the two streets according to the rule of four are: 12 * 4 = 48%.
- When calculating in the calculator, they will give out similar values - 25.53% and 44.96%.
These rules give results that practically do not differ from real ones. This helps make positive decisions and saves time at the table.
A small error appears when calculating a large number of outs. In this situation, you can use the 3 + 9 rule. If you need 10 outs on two streets to make a combination, the calculation would be 3 * 10 + 9 = 39%.
This value is very close to the real one – 38.4%.
When determining this indicator, it is important to assess the size of the bank and the size of the bet. They allow you to find out whether it is profitable to call an opponent’s raise or not.
Pot odds are the ratio of the bet a player can call to the size of the pot. By comparing them with the probability of winning, he understands whether calling is profitable.
What needs to be considered in the calculations:
- Money that is already in the bank after the last round of trading.
- Bets of all participants in this round, including their own.
- Bets to be placed by players after.
- Pot odds are calculated based on accurate information as the poker player knows the specific pot size and opponent’s bets.
By building the opponent’s hand range, the poker player can determine the size of his bet, which will be disadvantageous for his opponent in pot odds.
How to determine pot odds
Let’s look at an example. The pot on the turn is $ 15, after which Villain bets $ 5 and the pot is $ 20. To continue the game, the second participant needs to call $ 5.
To calculate the odds, you need to divide the pot size by the bet you want to call: $ 20 / $ 5 = 4: 1 Pot odds and this ratio will be 4 to 1.
As you can see from the example, the calculations do not depend on the board structure or starting hand. They are monetary indicators that reflect the share of investments from the possible gain. This data helps to determine if the bet will be profitable or not.
Comparison of odds
To determine the profitability of the bet, you need to compare the probability of falling out and the odds of the pot. If the pot odds are lower, calling or betting is profitable from a distance.
Example. On the flop, the player gets a straight draw, which requires any ace or any 9 to close for a total of 8 outs. Based on the table, the probability of appearing on the turn is 17.02% or 4.88: 1.
The pot is $ 50 and your opponent bets $ 25. Our pot odds are (50 + 25) / 25 = 3: 1.
It is believed that calling is beneficial when pot odds are more likely to gain. Since 3: 1 is less than 4.88: 1, it’s best to fold.
The rule is easily verified by mathematics. If in this situation every time you call, 1,000 / 4.88 = 204.92 pots of $ 75 will be won at a distance of 1,000 hands. The remaining 795.08 hands will have lost $ 25.
- Total won: 204.92 * 75 = $ 15,369
- Total Lost: 795.08 * 25 = $ 19.877
In this situation, at a distance of 1,000 hands, each time the player calls a bet, the poker player will lose $ 4,508. That is, on average $ 4.508 per hand. Therefore, the best solution is to reset.
Potential pot odds
In the examples above, draws were taken with a single opponent. However, the odds change if there are more streets ahead or another player makes a decision afterwards.
For this, the potential pot odds or implied odds are considered – these are odds, taking into account the additional bets that can be made in this and subsequent rounds. That is, the money that the poker player expects to win when hitting outs.
If opponents are willing to invest a lot on the next streets after the out, then there are good implied odds. If other players tend to fold, the implied odds are low.
It’s not just the strength of the hand that increases the potential odds. A poker player can exploit the play of opponents on the late streets if:
- Uses a profitable bluff.
- Villain does not fold to the continuation bet.
- Villain draws any draws to the river.
Therefore, a player can call without even having the right odds. However, good hand reading skills are required to correctly evaluate the implants.
How to Calculate Potential Pot Odds
Absolutely accurate calculation of implied odds is impossible due to taking into account a large number of variables: next board map, action, bet sizes, psychological state of the opponent. An experienced player should consider these components, but the best thing to do in this situation is to figure it out by eye.
An accurate calculation will only be in the situation when the player is in position against the opponents. In other cases, a lot depends on the ability to observe opponents and make correct assumptions about further actions.
However, you can calculate the minimum amount you need to win against your opponent on subsequent streets for a call to be profitable.
The implied odds can be calculated using the formula:
Implied Odds = Call / (Pot Size + Opponents’ Calls After Player + Estimated Winnings On Next Streets)
Example. We are in the big blind, 2 more opponents have entered the hand. The flop is $ 30. The small blind bet $ 10. Normally, pot odds are 4: 1, which is (10 + 30) / 10.
However, we are sure that the player sitting behind us on the button will also call the bet. The implies are (30 + 10 + 10) / 10 = 5: 1.
We risk $ 10 to win not $ 40, but at least $ 50. This means that for a profitable call, the equity of the hand must be at least 16.67%.
In addition to using ready-made tables and formulas for calculating odds, players can use additional programs.
An alternative is the use of poker calculators, which determine the probabilities automatically by reading data from history, or require manual input of cards for calculations.
The software allows you to analyze the situation, calculate probabilities, hand ranges, and choose the most profitable solution. Programs receive data from a distribution file or after manual user input and instantly perform calculations.
Thanks to calculators, you can see the probability of winning in percentages, indicating the player’s pocket cards, community cards in the distribution and the range of opponents’ hands.
However, you should not rely only on third-party software. You cannot use it during the game, so you need to consider common situations and memorize decisions against ranges and specific hands of opponents.
Studying poker odds and other mathematical aspects can greatly increase the knowledge, level of play and win rate of a poker player. Understanding poker probabilities and making decisions based on odds comparison allows you to win more often than opponents who rely on luck.