KEY TAKEAWAYS
Gambling is a language of chance, and probability is its grammar. Every payout, house edge and streak can be described with a small set of precise words: outcome, event, expected value, variance and a handful of others. This glossary defines 28 of those words in plain language, with a short example of each in use. The entries are organized into four groups: the building blocks of probability, expectation and the house edge, variation over time, and common misconceptions about randomness. Examples use standard figures that can be checked by hand, such as two dice, a 52-card deck and a roulette wheel. Terms already defined in the betting odds glossary, such as implied probability and overround, appear only as short cross-references. Nothing here is a betting system: no definition changes the house edge, and the aim is only to make the vocabulary of chance clear.
Building Blocks of Probability
These terms describe what can happen in a chance situation and how to count the possibilities.
Outcome
Definition: One possible result of a single trial of a chance process. Rolling a die has six outcomes, and a roulette spin on an American wheel has 38.
Example: “The outcome of that spin was 17.”
Related terms: sample space, event.
Sample space
Definition: The complete set of all possible outcomes of a chance process. For one roll of a die it is the numbers 1 through 6, and for two dice it contains 36 equally likely combinations.
Example: “The sample space for two dice has 36 outcomes, and six of them add up to 7.”
Related terms: outcome, event.
Event
Definition: A set of one or more outcomes that a bettor cares about, such as rolling an even number or drawing a heart. An event is said to occur if any outcome in it occurs.
Example: “The event ‘a total of 7 on two dice’ contains six of the 36 outcomes, so its probability is 6/36, about 16.67%.”
Related terms: outcome, complement.
Independent events
Definition: Events where one does not change the probability of the other. Successive coin flips, dice rolls and roulette spins are independent, because the equipment has no memory of earlier results.
Example: “Two roulette spins are independent, so the first result tells you nothing about the second.”
Related terms: dependent events, gambler’s fallacy.
Dependent events
Definition: Events where one result changes the probability of the next. Cards dealt from a deck without replacement are the usual example, because each card removed changes what is left.
Example: “The chance of a second ace is lower after the first ace has been dealt: 3 in 51 rather than 4 in 52.”
Related terms: independent events, conditional probability.
Mutually exclusive events
Definition: Events that cannot both happen on the same trial. A single die cannot show 2 and 5 at once, so the probability of 2 or 5 is the sum of the two: 2/6, about 33.33%.
Example: “A hand cannot be both a flush and a straight flush at the same time, so they are mutually exclusive categories.”
Related terms: event, complement.
Complement
Definition: The event that something does not happen. Its probability is 1 minus the probability of the event, which often makes ‘at least one’ questions easier to work out.
Example: “The probability of not rolling a 7 on two dice is 1 − 6/36 = 30/36, about 83.33%.”
Related terms: event.
Odds
Definition: A way of comparing the number of unfavorable outcomes with the number of favorable ones, written as a ratio such as 5 to 1. Odds of 5 to 1 against mean 5 unfavorable outcomes for every 1 favorable one, which is a probability of 1 ÷ (5 + 1), about 16.67%.
Example: “The odds against rolling a specific number on one die are 5 to 1.”
Odds and probability are not the same number. Related terms: probability and odds against, defined in the betting odds glossary.
Permutation
Definition: An arrangement of items where order matters. If three of eight horses must finish first, second and third in an exact order, there are 8 × 7 × 6 = 336 permutations.
Example: “There are 336 permutations of the top three finishers in an eight-horse field.”
Related terms: combination.
Combination
Definition: A selection of items where order does not matter. Choosing any 3 of 8 horses, in any order, gives 56 combinations. A five-card poker hand is a combination of 5 cards from 52, and there are 2,598,960 of them.
Example: “There are 2,598,960 possible five-card hands, and 4 of them are royal flushes, or 1 in 649,740.”
Related terms: permutation, sample space.
Expectation and the House Edge
These terms describe what a bet is worth on average, and how that average relates to what the house keeps.
Expected value
Definition: The average amount a bet wins or loses per unit wagered if it were repeated a very large number of times. It is found by multiplying each possible result by its probability and adding the results. It describes the long run, not any single bet.
Example: “A $1 straight-up bet on an American wheel has an expected value of −$0.0526, a loss of about 5.26 cents per dollar.”
Often shortened to EV. Related terms: house edge, negative expectation.
Positive expectation
Definition: A bet or situation whose expected value is above zero, so that the average result over many repetitions is a gain. It is often written +EV. Standard casino games are designed so that the player’s side does not have it.
Example: “A fair coin flip paid at 2 to 1 would be a positive expectation, but no standard casino bet is offered that way.”
Related terms: negative expectation.
Negative expectation
Definition: A bet whose expected value is below zero, so that the average result over many repetitions is a loss. Every standard bet on a casino game with a house edge is a negative expectation bet, even though individual bets can win.
Example: “Every bet on a European roulette wheel has a negative expectation of about 2.70%.”
Related terms: expected value, house edge.
House edge
Definition: The average share of each bet that the casino keeps over the long run, equal to the player’s negative expected value as a percentage of the amount wagered. It is built into the payouts.
Example: “The house edge is 5.26% on American roulette and 2.70% on European roulette.”
Related terms: return to player, expected value. House edge varies by game and rules, and is not changed by any betting system.
Return to player
Definition: The share of money wagered that a game pays back on average over the long run, equal to 100% minus the house edge. American roulette returns about 94.74% and European roulette about 97.30%. It is a long-run average across all players, not a promise to any one player.
Example: “A game with a 96% return to player keeps 4% of money wagered on average.”
Often shortened to RTP. Related terms: house edge, hit frequency.
Hit frequency
Definition: How often a bet or game produces a winning result, shown as a percentage of trials. It is separate from return to player: a game can win often in small amounts or rarely in large ones.
Example: “A slot that pays on 1 spin in 4 has a hit frequency of 25%.”
Related terms: return to player, volatility.
Break-even probability
Definition: The win probability at which a bet would have an expected value of zero, found by dividing the amount risked by the total returned on a win. A bet that pays 35 to 1 breaks even at 1 ÷ 36, about 2.78%, while an American wheel gives 1 ÷ 38, about 2.63%.
Example: “At 35 to 1, the break-even probability is about 2.78%, and the actual chance is slightly lower.”
Related terms: expected value, house edge.
Variation Over Time
These terms describe how far real results can stray from the average, and why short runs behave differently from long ones.
Variance
Definition: A measure of how widely the results of a bet are spread around its average. A bet with a rare, large payout has higher variance than an even-money bet with the same expected value.
Example: “Straight-up roulette bets have much higher variance than red or black bets.”
Related terms: standard deviation, volatility.
Standard deviation
Definition: The square root of the variance, expressed in the same units as the bet. It describes the typical size of the gap between a single result and the average. A $1 straight-up bet on an American wheel has a standard deviation of about $5.76 per spin.
Example: “The standard deviation of a single $1 even-money bet is about $1.”
Related terms: variance.
Volatility
Definition: A general word for how much results swing, usually used of slot machines and other games. High-volatility games tend to pay rarely but in larger amounts, and low-volatility games pay more often in smaller amounts.
Example: “The slot is described as high volatility, with long gaps between bigger wins.”
Register: informal. Related terms: variance, hit frequency.
Law of large numbers
Definition: The principle that the average result of many repeated independent trials moves toward the expected value as the number of trials grows. It does not mean past losses or wins are made up for in the short run.
Example: “Over 100,000 spins the average loss per spin will sit very close to 5.26% of the bet, but 100 spins can look quite different.”
Related terms: expected value, regression to the mean.
Regression to the mean
Definition: The tendency for an unusually high or low run of results to be followed by results closer to the average. It is a statistical pattern in samples, not a force that corrects an individual game.
Example: “After an unusually lucky week, results tend to look more ordinary, because the lucky run was partly chance.”
Related terms: law of large numbers, streak.
Risk of ruin
Definition: The probability that a bankroll is lost completely before a goal is reached or a stopping point arrives. It rises with larger bets relative to the bankroll and with higher variance. It is a measure of risk, not a way to remove it.
Example: “With a small bankroll and large bets, the risk of ruin is high even on a fair game.”
Related terms: variance. Money vocabulary such as bankroll is defined in the gambling money terms glossary.
Randomness and Common Misconceptions
These terms cover how random events behave in practice, and the mistaken beliefs that most often come from misunderstanding them.
Streak
Definition: A run of the same result in a row, such as five reds or four passes. Streaks are a normal feature of random sequences and do not signal that the next result is changed.
Example: “Five reds in a row has a probability of about 2.38% on an American wheel, and it still happens.”
Register: informal. Related terms: gambler’s fallacy, hot hand fallacy.
Gambler’s fallacy
Definition: The mistaken belief that a streak of one result makes the opposite result more likely next, even though the events are independent. A failure to recognize the independence of chance events is how the APA Dictionary of Psychology describes it.
Example: “After five reds in a row he said black was due, but the chance of red on the next spin was still 18 in 38, or 47.37%.”
Related terms: independent events, hot hand fallacy. On a European wheel the chance of red is 18 in 37, or 48.65%.
Hot hand fallacy
Definition: The belief that a person or machine that has just won is more likely to keep winning in a game of pure chance. In independent games such as dice, roulette and slots, a recent win does not change the next result.
Example: “He kept betting because the table was hot, but each roll of the dice was independent.”
Register: informal. Related terms: gambler’s fallacy, independent events. The idea of a hot hand in skill-based sports is a separate and debated question.
Conditional probability
Definition: The probability of an event given that another event has already happened. It is written P(A | B). It is the key idea behind dependent events such as cards dealt from a deck.
Example: “The probability of a second ace given that the first card was an ace is 3 in 51, about 5.88%.”
Related terms: dependent events.
Random number generator
Definition: A device or program that produces a sequence of results that cannot be predicted from earlier ones. Electronic slots and online casino games use them, and in many regulated markets they are required to be independently tested. Shortened to RNG.
Example: “The result of each spin is decided by the random number generator at the moment of the spin.”
Related terms: independent events, streak.
Terms Defined Elsewhere
These terms are defined once in the betting odds glossary. Each entry below is a cross-reference rather than a new definition.
Probability — see Betting Odds Terms: 24 Definitions With Examples.
Implied probability — see Betting Odds Terms: 24 Definitions With Examples.
Fair odds — see Betting Odds Terms: 24 Definitions With Examples.
Overround — see Betting Odds Terms: 24 Definitions With Examples.
Quick-Reference Table of Probability Terms
Use this table to find a term fast. Each term links to its full entry above.
| Term | Short definition |
|---|---|
| Outcome | One possible result of a single trial of a chance process. |
| Sample space | The complete set of all possible outcomes of a chance process. |
| Event | A set of one or more outcomes that a bettor cares about, such as rolling an even number or drawing a heart. |
| Independent events | Events where one does not change the probability of the other. |
| Dependent events | Events where one result changes the probability of the next. |
| Mutually exclusive events | Events that cannot both happen on the same trial. |
| Complement | The event that something does not happen. |
| Odds | A way of comparing the number of unfavorable outcomes with the number of favorable ones, written as a ratio such as 5 to 1. |
| Permutation | An arrangement of items where order matters. |
| Combination | A selection of items where order does not matter. |
| Expected value | The average amount a bet wins or loses per unit wagered if it were repeated a very large number of times. |
| Positive expectation | A bet or situation whose expected value is above zero, so that the average result over many repetitions is a gain. |
| Negative expectation | A bet whose expected value is below zero, so that the average result over many repetitions is a loss. |
| House edge | The average share of each bet that the casino keeps over the long run, equal to the player’s negative expected value as a percentage of the amount wagered. |
| Return to player | The share of money wagered that a game pays back on average over the long run, equal to 100% minus the house edge. |
| Hit frequency | How often a bet or game produces a winning result, shown as a percentage of trials. |
| Break-even probability | The win probability at which a bet would have an expected value of zero, found by dividing the amount risked by the total returned on a win. |
| Variance | A measure of how widely the results of a bet are spread around its average. |
| Standard deviation | The square root of the variance, expressed in the same units as the bet. |
| Volatility | A general word for how much results swing, usually used of slot machines and other games. |
| Law of large numbers | The principle that the average result of many repeated independent trials moves toward the expected value as the number of trials grows. |
| Regression to the mean | The tendency for an unusually high or low run of results to be followed by results closer to the average. |
| Risk of ruin | The probability that a bankroll is lost completely before a goal is reached or a stopping point arrives. |
| Streak | A run of the same result in a row, such as five reds or four passes. |
| Gambler’s fallacy | The mistaken belief that a streak of one result makes the opposite result more likely next, even though the events are independent. |
| Hot hand fallacy | The belief that a person or machine that has just won is more likely to keep winning in a game of pure chance. |
| Conditional probability | The probability of an event given that another event has already happened. |
| Random number generator | A device or program that produces a sequence of results that cannot be predicted from earlier ones. |
Related Glossaries
Odds formats and the terms for payouts and price are collected in Betting Odds Terms: 24 Definitions With Examples. Money words such as bankroll and stake are in Gambling Money Terms: Bankroll, Stake, Vig and More, and short forms such as EV and RTP are in Gambling Acronyms and Abbreviations: 24 Definitions. Dice combinations come up again in Craps Terms and Slang: 30 Definitions With Examples and Sic Bo Rules: Bets, Payouts and How the Game Works, and how sportsbooks use lines and juice is covered in Sportsbook Terms: 28 Definitions With Examples.
