When choosing a slot machine from the online casino collection, first of all, you need to pay attention to the number of symbols and reels. In accordance with the theory of probability, the key principle can be formulated as follows – the more symbols and reels, the lower the chances, and vice versa.

The first step in calculating the probability of winning is calculating the possible number of combinations.

To win three-reel classic slots, the player needs to match three identical symbols on one line. It is necessary to proceed from the assumption that symbols are not repeated on each reel, that is, 1 reel = 1 winning symbol. This assumption simplifies calculations and in most cases corresponds to reality.

To calculate the number of combinations, you just need to multiply the number of symbols on each reel by itself only once, how many reels are in the slot. If we are talking about a three-reel online slot with twenty symbols on each of them, then you will need to perform the following mathematical action: 20x20x20 = 8000.

That is, the gaming machine is capable of giving out eight thousand different combinations of symbols. The player needs one winning one. That is, the chance of winning will be 1/8000 or 0.0125%. The figure seems small, but not in comparison with the probability of a combination falling out on a five-reel model with the same number of symbols on the reel – 0.0003125%, that is, several orders of magnitude less.

What Strategy Should You Use?

Experienced gamblers usually advise using the “Empty Spins” scheme when playing classic emulators of mechanical slot machines. This strategic scheme does not in any way affect the probability of getting a winning combination, but it effectively helps to control the gambler’s expenses on gambling. The essence of the strategy boils down to the fact that the player sets for himself a certain number of losing spins. As soon as the number of unsuccessful spins reaches the established one, it is imperative to change the slot machine or completely end the game and return to the club not earlier than the next day.

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.

Outs 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.

Probabilities 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

one

2.10%

2.10%

4.20%

2

4.20%

4.20%

8.40%

3

6.40%

6.50%

12.50%

four

8.50%

8.70%

16.50%

five

10.60%

10.90%

20.40%

6

12.80%

13%

24.18%

7

14.90%

15.20%

27.80%

eight

17%

17.40%

31.50%

nine

19.10%

19.60%

35%

10

21.20%

21.50%

38.40%

eleven

23.40

23.90%

41.70%

12

25.50%

26%

45.00%

13

27.70%

28.20%

48.10%

fourteen

29.80%

30.40%

51.20%

fifteen

31.90%

32.60%

54.10%

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

Poker odds table for frequent hands and draws

Situation

Example

Odds

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%

Equity in common situations

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.

Relationship method

For calculations in the form of relations, you can use the formula:

Odds = Inappropriate cards / Cards that will improve the combo

Or

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.

Percentage method

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%.

Pot odds

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.

Result:

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%.

Automatic calculation

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.

Now you know all the basic stakes in roulette and payout ratios for them. So let’s move on to the most interesting part of this article. It’s time to count the odds in roulette to find out the value of each bet in the long run.

I calculated the so-called RTP (Return to Player) for each bet in the table below. RTP is the percentage of payments . This indicator indicates the value of the bet.

Domestic betting: chances and probabilities

All the information you need about the odds and probabilities of internal bets is given in the table below. Some bets are only available in the American or European versions of roulette, so some table cells are left blank.

Rate

Number of rooms

Payout ratio

Chance to win (Heb.)

Chance to win (Amer.)

RTP rates (Heb.)

RTP rates (Amer.)

Straight app

1

36: 1

1/37

1/38

97.3%

94.74%

Split

2

18: 1

2/37

2/38

97.3%

94.74%

Street

3

12: 1

3/37

3/38

97.3%

94.74%

Line

6

6: 1

6/37

6/38

97.3%

94.74%

Angle

4

9: 1

4/37

4/38

97.3%

94.74%

Trio

3

12: 1

3/37

–

97.3%

–

Basketball

4

9: 1

4/37

–

97.3%

–

Top line

5

7: 1

–

5/38

–

92.1%

Note that the payout percentage of the top line in American roulette is lower than the payout percentage of all other bets. This is all because the rate covers 5 numbers and the payout ratio is only 7: 1. If the ratio was 7.2: 1, the percentage of payments at this rate would be equal to others in American roulette. That’s why a top line bet will never be a good idea .

External bets: chances and probabilities

In the table below, I have provided you with all the information you need about the odds and probabilities of external betting .

Rate

RTP (Hebrew)

RTP (Hebrew + “La Partage”)

RTP (Hebrew + “En Prison”)

RTP (Amer.)

RTP (American + Surrender)

Red / black

97.3%

98.6%

98.6%

94.74%

97.2%

Even / odd

97.3%

98.6%

98.6%

94.74%

97.2%

Large / small

97.3%

98.6%

98.6%

94.74%

97.2%

Due to these rules, odds at the level of odds have a higher payout percentage compared to other bets . Therefore, if you play roulette where one of the above rules applies, remember that these bets will generally be more profitable.

Oral bets: chances and probabilities

Oral rates are special. They consist of smaller bets covering certain sectors of the roulette board. This means that the odds and probabilities of verbal bets can be divided by the odds and probabilities of each of the lower bets .

Therefore, I will not list the payout percentage of these rates, as I did for standard internal and external rates. Any oral bet on roulette with one zero has an RTP of 97.3% . Oral bets consist of internal bets, which means that the payout percentage of the total bet will not change when special rules are applied that affect the payout percentage of the odds level.

Oral bets are almost always used only in European (French) roulette. However, when used (as modified) in American roulette, they would have a payout percentage of 94.74% , provided they do not include a top-line bet, as it has a slightly different RTP and can change it for the overall bet.

Percentage advantage of casino in roulette (share of casino) The fate of the casino is the overall percentage advantage of the casino . Calculate it for any game is quite simple:

Casino share = 100% – RTP (payout percentage)

Accordingly, here is the standard share of a casino for roulette:

European roulette (roulette with one zero) – 2.7%;

American roulette (roulette with double-zero) – 5.26%.

However, as you probably already understand, there are some exceptions : they are some bets and special roulette rules, as they can reduce the advantage of the casino over the players.

The following are the rates and rules that can change the fate of the casino:

La Partage and En Prison rules reduce the share of a single zero roulette casino for odds (red / black, even / odd, big / small) to 1.35% ;

the “Surrender” rule reduces the share of casino roulette with double-zero for betting at the level of odds (red / black, even / odd /, big / small) to 2.6% ;

the top line bet, which covers numbers 0, 00, 1, 2 and 3 on a double-zero roulette, has the highest percentage advantage of the casino among all – 7.9% .

The lower the share of the casino, the better for the players . In terms of the payout percentage and the casino’s interest advantage, the odds on equal odds in French roulette with the “La Partage” or “En Prison” rule are considered the best for players, as they statistically lose only 1.35% of their bets.

What is the fate of casino roulette compared to other casino games?

Due to the share of casinos and the percentage of payouts, it is quite simple to compare different casino games and determine which of them are the least profitable casinos and the most profitable for players. So, will the fate of the casino in roulette withstand comparison with other casino games?

Below are the percentage benefits of the casino for different games.

European roulette – 2.7%.

American roulette – 5.26%.

French roulette (with the rule “La Partage” or “En Prison” – 1.35%.

Blackjack (according to the main strategy ) – less than 0.5%.

Slots – from about 1% to 8% (average 4%).

Baccarat – about 1% (when betting on the bank).

Video Poker (Valts and Better) – less than 0.5%.

As you can see, everything is recognizable by comparison, because there are quite a few games that are more profitable for the player in the long run. American roulette is much worse for the player, so if you are going to play roulette, find a casino where there will be a roulette with one zero and, ideally, using rules that reduce the casino’s percentage advantage.

How to calculate RTP and the share of casinos in roulette?

As already mentioned, roulette is a fairly simple game, so calculating the percentage of payouts and the share of the casino is not difficult. In this part of the article, I will explain why casinos always have an advantage in roulette , as this advantage is mathematically related to the course of the game and the chances of winning.

For calculations I use European roulette without special rules. Although this formula is suitable for other types of roulette, if you make minor changes to the benefits of the casino in each case.

Mathematical expectation or expected value ($ 1 bet on straight up) = Possible win * Odds of winning = $ 36 * (1/37) = $ 36/37 = $ 0.973

For other bets, the expected value will be the same.

$ 1 in red / black = $ 2 * (18/37) = $ 36/37 = $ 0.973

I could do this calculation for each type of bet in European roulette, but is it really necessary? Payouts for each bet are calculated according to the same formula, which means that each European roulette has a payout percentage of 97.3% (statistically, a player wins $ 0.973 for every $ 1 invested).

The payout percentage in European roulette is the same for all types of bets. However, if special rules are applied that may make certain bets less profitable for the casino, the payout percentage increases and the casino share decreases.

Calculations for American roulette are quite similar. All you have to do is replace the number “37” with “38”, because there are 38 possible game results in a double-zero roulette.

Each result in American roulette is slightly less likely than in Europe (except for “00”, which is not on roulette with one zero), while the payouts remain the same. That is why American roulette has a lower payout percentage – only 94.74% .

And one bet in American roulette has an even lower payout percentage – 92.1%. The reason for this is the non-standard chances of winning (5:38) and the payout ratio (7: 1) for the “top line” bet. The RTP calculation for this bet is as follows: $ 1 on top line = $ 7 * (5/38) = $ 35/38 = $ 0.921

The share of the casino can be calculated using the payout percentage:

Casino share = 1 – RTP

Accordingly, the fate of the casino in roulette is as follows:

1 – 97.3% = 2.7% for European roulette;

1 – 94.74% = 5.26% for American roulette.

Of all the casino games, roulette does not offer the highest chances of winning, and the games are more exciting. However, there are many who want to argue that roulette is a classic among casino board games that brings a lot of fun.

Moreover, the rules of the game are very simple, and the chances of winning are quite high . It can be played even by beginners who know absolutely nothing about roulette, although I recommend that you first learn the basics and rules of the game to make only the right decisions.

That’s why I decided to write this article. I hope it will be useful to you, and I wish you success at the roulette table if you decide to try your luck in this game.