18 
- Probabilities as the Art of Prediction
- Pot Odds. When Risks are Justified
- Expected Value (EV): Assessing Long-Term Profits
- Long-Term Strategy, Long-Term Play
- How Technology Helps to Use Mathematics
- Poker Mathematics in Practice
- How to Win in Poker
- FAQ
- What are pot odds?
- How to calculate the probability of making the desired hand?
- What is expected value (EV) in poker?
- Why is it important to play for the long run?
- How does technology help improve mathematical calculations?
- Can you use pot odds without knowing probabilities?
At the poker table, every decision matters: whether to invest more chips, continue the hand or fold. Many people think of poker as a game of intuition and luck, but true professionals know that there is science behind their success — the mathematics of poker. It is the calculations that help turn chaos into a clear strategy and find optimal solutions even in the most difficult situations. Let’s look at the key mathematical concepts that turn an amateur into a professional: probabilities, pot odds, and expected value. You will learn how these tools work in practice, and understand how using modern software makes your game even more accurate.
Probabilities as the Art of Prediction
Poker is a game of incomplete information. You don’t know the opponent’s holding or what will happen next, but you can calculate the probability of certain events. It is probabilities in poker that allow you to make thoughtful decisions rather than rely on sheer luck.
Chances of Improving Your Hand
Let’s say you have two clubs in your hand, and two more clubs are on the flop. To make a flush, you need one more card of this suit. There are 13 − 4 = 9 clubs left in the deck, and there are 47 cards left in the deck (52 minus your 2 and 3 on the board). This means that the probability of drawing the desired card on the turn will be:
9 / 47 ≈ 19%
If the flush is not completed on the turn, the odds of getting it on the river are different. There are only 46 cards left in the deck, and the probability of hitting the desired suit increases to 9 / 46 ≈ 20%.
Probabilities and Pot Odds
Understanding probabilities becomes even more useful if you combine them with pot odds (more on that later). For example, if the probability of making a flush is 19%, and to continue the game you need to call a bet that is less than 19% of the pot, the call is justified.
Pot Odds. When Risks are Justified
Pot odds are the ratio between the current pot and your bet, which helps you decide whether to continue playing. Players often refer to this as “buying a card.” The larger the pot relative to the bet, the more likely it is that it is worthwhile to continue playing, even with a questionable hand.
How it works
Let’s look at a simple example: there is $100 in the pot, and your opponent bets $25. To continue playing, you need to put in $25. Your pot odds are:
Pot odds = (pot + opponent’s bet) : your bet = ($100 + $25) : $25 = 125:25 = 5:1
If the probability of winning with your hand is higher than the pot odds (in this case, 20%), the call is justified.
Long-Term Odds
Pot odds are especially important for players who think long-term. One wrong decision may seem profitable in the short term, but in the long term, it will lead to the empty bankroll.
Expected Value (EV): Assessing Long-Term Profits
Expected Value (EV) is a measure that allows you to estimate how profitable your decision will be in the long run. EV shows how much you will earn or lose if you repeat the same action over and over again.
How EV is calculated
EV is calculated as the difference between the expected profit and the expected losses:
- EV = (Probability of winning × Possible gain) − (Probability of losing × Possible loss)
Let’s say you are playing in a $200 pot and need to call $50 to continue. The probability of getting a winning combination is 30%. Then your EV will be:
- EV = (0.3 × $200) − (0.7 × $50) = $60 − $35 = $25
A positive EV ($25) means that this decision is profitable in the long run.
Negative EV
If the EV is negative, the decision is not worth making. For example, if the probability of winning is only 15%, then:
- EV = (0.15 × $200) − (0.85 × $50) = $30 − $42.5 = −$12.5
This means that every time you make this decision, you will lose money.
Long-Term Strategy, Long-Term Play
Understanding probabilities, pot odds, and EV helps you avoid the traps of short-term thinking. Poker is all about long-term play — the thousands of hands you play over a year.
Why it’s important to play long-term strategies:
- The EV advantage becomes obvious. Even if your hands don’t work out for a while, a positive EV pays off.
- Reducing the influence of luck. The more hands you play, the less variance plays a role.
- Bankroll management. Proper financial management helps you avoid going broke even during downswings.
How Technology Helps to Use Mathematics
Players are increasingly turning to services that analyze the game and help make decisions. One such service, HisHands, collects data on opponents and offers strategic tips.
HisHands collects statistics on your opponents and helps you get data such as:
- Betting frequencies.
- Bluffing tendencies.
- General playing style (tight, aggressive, etc.).
This data helps you more accurately assess probabilities and make decisions based on real statistics, not assumptions. At the same time, the service collects data on each player during all games, so even at a new table you will have an undeniable advantage.
Poker Mathematics in Practice
Let’s look at an example of a real game situation where mathematics helps to make the right decision.
Situation:
You have J♦Q♦, and the flop comes K♦10♦7♠. You have a draw to a straight and a flush. The pot is $150, the opponent bets $50. What to do?
1. Calculate the probabilities:
- You have 9 outs to a flush and 6 outs to a straight (3 nines and 3 aces), a total of 15 outs.
- Probability on the turn = 15 / 47 ≈ 32%.
2. Calculate the pot odds:
- Current pot = $150 + $50 = $200.
- Bet size = $50.
- Pot odds = $200 : $50 = 4:1 (25%).
3. Compare the probabilities with the pot odds:
- The probability of making a desired hand (32%) is higher than the pot odds (25%). This justifies the call.
How to Win in Poker
Poker math is an essential tool for anyone who wants to play at a professional level. Understanding probabilities, pot odds, and EV allows you to make optimal decisions at the table and minimize the influence of luck. And modern services like HisHands turn data into the key to success, helping to analyze the game and improve the accuracy of calculations.
Poker is not just a card game, but also an intellectual challenge. The deeper you understand the mathematics of the game, the higher your chances of winning.
FAQ
What are pot odds?
Pot odds are the ratio between the current pot size and the bet size you need to call to continue playing. They help determine whether a call is mathematically justified. If the probability of winning your hand is higher than the pot odds, then it is profitable to continue playing.
How to calculate the probability of making the desired hand?
To calculate the probability, you need to determine the number of outs – cards that improve your hand. For example, if you have a flush draw (four cards of the same suit), then there are 9 cards of this suit left in the deck. Next, divide the number of outs by the number of remaining unknown cards. For example, the probability of improving the flush on the turn = 9 outs / 47 unknown cards ≈ 19%.
What is expected value (EV) in poker?
Mathematical expectation (Expected Value, EV) is an indicator that estimates how profitable your decision will be in the long run. EV is calculated as the difference between the expected profit and the costs. If the EV is positive, the decision is mathematically profitable.
Why is it important to play for the long run?
In the short run, poker is heavily dependent on luck. However, in the long run (hundreds and thousands of hands), your mathematically sound decisions determine the result. Even temporary failures pay off if you make positive EV decisions.
How does technology help improve mathematical calculations?
Services like HisHands collect data on opponents and their playing style, which allows you to more accurately calculate probabilities and make informed decisions. This is especially useful if you do not have enough experience to analyze the situation on your own.
Can you use pot odds without knowing probabilities?
Pot odds and probabilities work in combination. Even a simple understanding of pot odds can help you make a better decision, but without calculating probabilities, you will not be able to assess the exact risks. Therefore, a basic knowledge of both concepts is necessary for successful play.
