Key statistical terms in gambling – definitions and adequate interpretations

Written by

Dr. Catalin Barboianu

Oct 30, 2025

15 min read

Updated: Nov 3, 2025

Games of chance are conceived on the basis of mathematical models, which ensure their functionality and guarantee profit for their operators in the long run. This is why the complete descriptions of these games, as well as their associated economic and profitability indicators use terms of a mathematical-statistical nature. These terms also characterize gambling strategies and quantify the gambling activity.

Statistical terms used in gambling concern the gambling descriptions and gambling discourse for all parties involved in the phenomenon – experts, operators, regulators, and players. However, given their statistical nature, their usage in real-life gambling contexts is many times inadequate and can be misleading.

In this article you will see with examples what the key statistical gambling terms are and how should we interpret them adequately in a gambling.

Odds as probability

‘Odds’ is the most used technical term in gambling. However, in gambling jargon ‘odds’ may have several meanings and its undifferentiated usage may generate confusion and may be misleading.

The main meaning of ‘odds’ is that of mathematical probability. Probability is a mathematical measure of the possibility of an event happening, provided that the event belongs to a certain mathematical structure (called sigma field). As a measure, probability obey some axioms, namely to take values in the interval [0, 1], taking value 1 for the entire sample space (the sure event), and to be additive, that is, the probability of two mutually exclusive events is the sum of their probabilities.

In the simplest terms, probability of an event is the ratio between the number of situations or evidences favorable for that event to occur and the total number of situations or evidences, which form the sample space. This is called classical (Laplacian) probability and applies in gambling to all the events related to games’ outcomes, which belong to a finite sample space.

Quick Probability Examples (Dice)

1. The probability for a die to roll a 5 is 1/6 as a fraction, or 16.66% as percent, because there is one situation favorable for that event (the die showing the face with no. 5) out of six possible situations.
2. The probability for two die to roll a sum of 6 is 5/36 or 13.88%, because there are 36 possible outcomes (6 for each die), of which 5 are favorable for that event, namely the ordered pairs (1, 5), (2, 4), (3, 3), (4, 2), (5, 1).

Combinatorics Example: Two Aces in Texas Hold’em

In certain games, the outcomes are combinations of elements, like in card games such as blackjack, baccarat, or poker (cards), and slots (symbols). To calculate probabilities of events expressed by combinations, certain combinatorial computations are necessary before applying the classical definition of probability.

For computing the probability of one being dealt two aces as hole cards in a Hold’em poker game, we count first the number of combinations of two aces, which is C(4,2) = 6, then the total number of possibilities of dealing two cards, which is C(52,2) = 1326. Now we do the division: 6/1326 = 0.00452 = 0.452% is the sought probability.

Probability computations are not always that simple as in our examples. More complex events in gambling, especially in blackjack and poker, require special mathematical techniques of approximation and some probabilities are not even computable by compact formulas, but only by computer algorithms or simulations.

Numerical probability can be expressed as a fraction, percentage (as in the above examples), but also in the so-called ‘odds format’, as relative to the probability of the contrary event. The following formula is used to convert probability (as fraction or percentage) into odds: odds = probability/(1 – probability)

Converting Probability to Betting Odds

For instance, A 1/4 probability of an event E is converted into odds as follows:

Odds of E = (1/4)/[1 – (1/4)] = (1/4)/(3/4) = 1/3. This is denoted by 3 : 1 or ‘3 to 1’ and reads as three against one, that is, the chances for E to occur are 3 against 1, meaning that there are 3 chances out of 4 for the opposite of E to occur and one chance in 4 for E to occur.

The odds format of probability is more specific to betting, but the odds format is usually used to also express the payout rates in most of the casino games, such as roulette, blackjack, baccarat or craps. This sharing of the odds format in two different contexts is many times a source of confusion. This is why it is recommended to use the term ‘probability’ when referring to mathematical probability and ‘payout odds’ or ‘payout rates’ when referring to what a game pays out at a win relative to the wager.

Probability (odds) is a mathematical notion expressing a measure, a kind of average or a limit for the possibility of an event happening. It does not provide any certainty about the occurrence of that event in a given setup or over a definite period. We have to always make the difference between probability of an event and its physical possibility of occurrence. The only relationship between probability as an abstract notion and the occurrences of real-life events is still mathematical, called the Law of Large Numbers, saying that the sequence of the relative frequencies of the occurrence of an event approaches its probability.

Odds as payout rate

The payout rate of a winning outcome in a game is the multiplier that applies to the wager to determine the payback (payout) for the winner. The payout rate is part of the rules of a game, which has a payout schedule where all winning outcomes are listed along with their payout rate.

The payout rate may be denoted as a multiplier (x…) or in odds format (… : … or … to …). The multiplier denotation is mostly used in slot games, and the odds format in all casino table games. The payout rate is sometimes called payout multiplier.

Payout Odds vs. Multipliers: Examples by Game

1. The payout rate of a street bet in roulette is 11 : 1 (or x11), meaning that if you win that bet you are paid back 11 times your wager and you are returned your wager.
2. In classical blackjack, there is a 3 : 2 (or x1.5) payout rate for a blackjack beating the dealer. This means that you are returned your wager and paid 1.5 times your wager.
3. In a slot game, if a winning combination has the payout rate x500, it means that you are paid back 500 times your credit (wager), without being returned your wager in addition.

Hence in casino table games the payout rates as displayed in the set of rules reflect the net gain when applied to the wager, while in slots they reflect a gross gain and you have to subtract one unit from the multiplier to get the net rate. The same gross gain is reflected by the payout rates in sports betting, which can be expressed in three possible formats, depending on geographic zone: decimal, fractional, or moneyline.

The payout rate is not actually any statistical notion or term, but it is an important parameter characterizing bets in a game, which is employed in most of the key statistical notions specific to gambling (such as expected value, house edge, RTP, etc.). We have to make the distinction between payout rate and payout. While payout rate expresses a fixed rate to be applied to the wager, payout should be meant in terms of actual amount to be paid back, depending on the wager.

In gambling jargon, but also in many expert descriptions, the term ‘odds’ is used to reflect the payout rate of a bet. In sports betting, it is an established language standard to use ‘odds’ with the meaning of payout rate. This does not create much confusion, however it does when used in the same context with ‘odds as probability’ and this stands for the discourse about any game.

In particular in sports betting, such confusion may be potentiated by the existence of the notion ‘implied probability’, which actually is not any mathematical probability, but just another way to express the payout rate of a bet (as a percentage). In gambling jargon, the distinction between the two meanings is best caught with the terms ‘true odds’ versus ‘payout odds’.

Therefore we have to careful when using or reading in given contexts the term ‘odds’ and we should not assign it interchangeable meanings, as there are different concepts expressed by the same word. For instance, if one consider the question “What game offers the best odds?”, different people may answer or discuss it in terms of either probability of winning, payout rate, or even house advantage as different meanings for ‘odds’.

Expected value

The expected value (or mathematical expectation) of a bet is an essential notion in gambling theory. It is a statistical indicator of a bet, defined in general as follows:

Expected Value (EV): Definition and Core Formulas

EV = (probability of winning) x (profit if you win) + (probability of losing) x (loss if you loose), where the loss is expressed as a negative number.

The payout rate is employed in the formula of the EV, as the profit depends on it. More precisely:

If S is the stake of the bet, p is the probability of winning it, and r is the net payout rate for the win, then: EV = p x r x S – (1 – p) x S

The expected value can also be expressed as percentage of the wager:
EV (%) = p x r – (1 – p)

The EV is applicable in any game, as any round consists of bets placed. In roulette, you bet on numbers or groups of numbers, in blackjack you bet that you will beat the dealer, in slots you bet that you will hit a winning combination, and so on.

Worked EV Examples: Roulette and Baccarat

1. In European roulette, a column bet pays out 2 to 1 and have a probability of winning of 12/37. Its expected value is EV(%) = (12/37) x 2 – [1 – (12/37)] = –1/37 = –2.70%. By running this bet indefinitely may times, one should expect to loose in average 2.70 cents at every dollar wagered.
2. In 8-deck baccarat, a Banker bet pays out 1 to 1 and has a probability of winning of 45.86%. Its expected value is EV(%) = (45.86/100) x 1 – [1 – (45.86/100)] = –8.28%. By running this bet indefinitely may times, one should expect to loose in average 8.28 cents at every dollar wagered.

In statistical terms, the expected value is a mean of a random variable, and should be understood not as an arithmetical mean, but as a weighted mean, where the weights are probabilities. From this description, it follows that the expected value is statistical average and should be interpreted as such. The expected value of a bet should not be interpreted in reality as a predictor for the gain/loss over a definite period of time or playing session or number of plays, but as an overall gain or loss specific for than bet in the ideal condition of placing that bet infinitely many times, or, in a more non-mathematical wording, over “the long run”.

How to Interpret EV in Real Play

Rolling a die for 12 times does not necessarily result in showing a 5 twice. Like probability of an event does not predict a frequency of any kind of its occurrences, but does stand for a limit (at infinity) expressed as an average, so does the expected value of a bet stand for the statistical average of the gain or loss.

As in our examples above, the expected value is negative for most of the gambling situations, indicating a loss for the player. It is so because the payout odds are so chosen in any game’s rules to favor the house, that is, the house should make a profit with that bet over the long run in whatever conditions.

There are a few exceptions where the expected value can become positive for some bets in some circumstances, in intermediary stages of a game, namely in conditions of optimal play. However, these exceptions do not affect the house’s guarantee for overall profit with that game over the long run.

The expected value is the most important statistical indicator in gambling, as it grounds other important statistical notions such as house edge, variance/volatility, and standard deviation. These statistical indicators are employed in games’ design for them to produce the results desired by their producer.

For the players, the expected value of a bet is a criterion employed in objective strategies, including optimal, as playing to reach the highest possible expected value of a bet by strategic moves and choices (in the games allowing them) means maximizing the gain and minimizing the loss in the long run.

House edge

Mathematically, the house edge (or house advantage) of a bet is defined as the opposite in sign of the expected value of that bet: HE = –EV (%). Hence, if the EV is negative, the HE should be positive. The immediate interpretation of the house edge is that it reflects the share of the wagers placed with a bet, which the house retains as their profit in the long run.

House Edge Formula and What It Means

For a formula of the HE of a bet, we have just to change every sign in the formula of the EV: HE = –p x r + 1 – p = 1 – p x (r + 1)

We can define the house edge of a game if there is only one type of bet in that game, even though the game may have several payouts for the various outcomes, that is, that game is a bet itself, not changing with the stages of the game.

In roulette, the payout rates are such chosen for any simple or combined bet to have the same expected value and implicitly the same house edge. Hence, the house edge of roulette is the same as the house edge of any of its bets (2.70% for European and 5.26% for American roulette).

House Edge by Game: Quick Reference

In games with multiple payout rates for various outcomes, for the same bet, such as blackjack or slots, all these rates are taken into account when computing house edge.

Example: Computing House Edge in Blackjack

In classical 2-deck blackjack, we have the following payout rates and apriori probabilities (before the game starts, when no cards are dealt), for the possible outcomes:

• Player wins with blackjack: ;
• Player wins without blackjack: ;
• Tie (push): ;
• Player looses: ;
• HE = –0.0455 x 1.5 – 0.3757 x 1 – 0.8680 x 0 – 0.4920 x (–1) = 0.0480 = 4.80%

In games like craps, there is no house edge as defined above. That’s because there are some bets (the place bet, for example) possibly requiring many rolls to resolve it. During these rolls, the player may cancel the bet at any time. In this case, there are three options to define the HE for a craps bet, namely per bet made, per bet resolved, and per roll.

It is important to know that the house edge of a game varies with the versions of that game (since the rules, payout rates, and probabilities change) and also with the player’s optimal strategies (for games allowing such strategies). For instance, in blackjack the house edge can reach 0.1% as its bottom value if the game is played with a card-counting strategy.

The value of the house edge being positive reflects the guarantee that the operator will always make profit with that game in the long run, regardless of any strategies the players may use for winning. As the expected value, the house edge is a statistical average itself and should be interpreted as such. It does not reflect the house’s profit as percentage of the wagers over a definite period of playing or session of a game, but an average theoretical percentage specific to an ideal endless functioning of the game.

House edge is an important statistical indicator of the games from a commercial standpoint, concerning the operators, but also from a strategic standpoint, as an objective criterion that players may use for choosing between games.

Return to player (RTP)

The return to player (or payback percentage), abbreviated as RTP, is a statistical indicator of a bet or game expressing the average share from the players’ wagers that are returned to them as prizes/wins in the long run.

RTP Explained: Relation to House Edge and EV

Mathematically, the RTP is just another form of expressing the house edge: RTP = 1 – HE = 1 + EV. Therefore, the higher the house edge, the lower the RTP.

For games with multiple payout rates for the same bet, RTP = 1 + p₁ × r₁ + p₂ × r₂ + … + pₙ × rₙ

The simplest way to express the RTP in general is RTP = (average win/average bet) × 100%, where the average win = prize × probability (of that prize).

RTP Examples by Game

1. The RTP in American roulette is RTP = 1 – 5.26% = 100% – 5.26% = 94.74%
2. Let’s calculate the RTP of a Tie bet in 8-deck baccarat: First, we have to calculate the EV:

The possible outcomes, along with their payout rates and probabilities, are as follows:

• Banker wins: r₁ = −1 ; p₁ = 0.4585
• Player wins: r₂ = −1 ; p₂ = 0.4462
• Tie wins: r₃ = 8 ; p₃ = 0.0951
• EV = 0.4585 x (–1) + 0.4462 x (–1) + 0.0951 x 8 = –0.1439.
• HE = 1 + EV = 0.8561 = 85.61%

RTP Components: Baccarat Tie Bet Breakdown

As a function of HE (or EV), the return to player is also a statistical average and is used mostly in the technical descriptions of the slot games.

This statistical nature of the RTP as a statistical average should be used to correct the various misconceptions and fallacies that the players may manifest about this notion (especially slot players):

• Of whatever value, the RTP does not reflect any kind of gain, but a loss.
• The RTP of a game should not be interpreted as the return for a given player from their own wager or wagers, but cumulatively, that is, the return from all players’ wagers to all players over the long run. Or, interpreting the RTP for just one player, it is the return from that player’s wagers to that player if they played that game an infinite number of times.

Conclusion

Key statistical terms in gambling such as odds and probability, expected value, house edge, and return to player are part of the technical descriptions of the games and indicators or criteria for both the experts and players, concerning the production of games, functioning and profitability analysis, and playing strategies.

Using these mathematical terms in non-mathematical contexts with unclear meanings may be sometimes conflicting and misleading. In addition, the probability theory notions may be tricky for those unfamiliar with them, especially when applied in the real life.

In order to have an adequate interpretation of these terms in both mathematical and physical contexts, we should know about both their mathematical definition and how they apply in the real life of gambling. This is actually a prerequisite of playing informed and avoiding misconceptions and fallacies, so common in gambling.