Dice Probability Calculator

Understanding the mathematical principles behind rolling dice is the cornerstone of probability theory, game design, and statistical risk assessment. By mastering these concepts, individuals can mathematically predict outcomes, balance complex game mechanics, and make perfectly informed decisions in scenarios governed by random chance. This comprehensive guide explores the exact mathematical formulas, historical origins, and practical applications of dice probability, equipping you with the knowledge to calculate the exact odds of any dice-based scenario.

What It Is and Why It Matters

Dice probability is a specialized branch of discrete mathematics and statistics dedicated to determining the exact mathematical likelihood of specific outcomes when rolling one or more polyhedral dice. At its core, it is the study of sample spaces—the complete set of all possible results a random event can produce—and the precise calculation of how often a specific target event will occur within that space. Because dice operate on a system of discrete uniform distribution, meaning every individual face of a perfectly balanced die has an exactly equal chance of landing face-up, they provide one of the purest and most accessible models for understanding probability theory. This predictability allows mathematicians, game designers, and statisticians to map out the exact mathematical boundaries of luck and chance.

Understanding this mathematical framework matters profoundly because it transforms chaotic randomness into predictable, manageable data. For tabletop roleplaying game (RPG) players and board gamers, understanding these probabilities dictates optimal strategic choices, turning blind guesses into calculated tactical decisions. For game designers, mastering dice probability is the fundamental requirement for creating balanced, engaging mechanics; a game where success is too mathematically improbable becomes frustrating, while a game where success is mathematically guaranteed becomes boring. Furthermore, in the realms of finance, actuarial science, and risk management, the mathematical models originally developed to understand dice rolls are the exact same models used to calculate insurance premiums, model stock market fluctuations, and predict the likelihood of catastrophic real-world events. Ultimately, calculating dice probability is the practice of quantifying the unknown, allowing human beings to make rational, mathematically sound decisions in a universe governed by unpredictable variables.

History and Origin of Dice and Probability

The history of calculating dice probability is inexorably linked to the birth of modern probability theory itself, a mathematical revolution that fundamentally changed how humanity understands the world. Long before mathematics was involved, humans used astragali—the asymmetrical ankle bones of sheep or goats—for both gambling and divination as early as 3000 BCE in Mesopotamia and Egypt. Early polyhedral dice, carved from bone, ivory, and clay, have been discovered in the ruins of ancient Sumer and the Indus Valley. However, for millennia, the outcomes of these rolls were widely believed to be determined by the will of the gods or pure fate, rather than by any underlying mathematical laws. The concept that chance could be quantified, measured, and predicted was entirely absent from human thought until the Renaissance.

The mathematical study of dice began in earnest with the Italian polymath Gerolamo Cardano in the 16th century. An avid gambler, Cardano wrote Liber de Ludo Aleae (The Book on Games of Chance) around 1564, though it was not published until 1663, long after his death. In this groundbreaking manuscript, Cardano defined the concept of the sample space for the first time, explicitly calculating the odds of rolling specific sums with two and three six-sided dice. He recognized that while individual rolls are unpredictable, the aggregate results over time strictly adhere to a mathematical ratio. This was the first time in recorded history that a human being formally articulated the mathematical laws governing random events.

The true watershed moment for dice probability, however, occurred in the year 1654 in France. A prominent writer and amateur mathematician named Antoine Gombaud, who styled himself the Chevalier de Méré, approached the brilliant mathematician Blaise Pascal with a gambling dilemma known as the "Problem of Points." The problem asked how to fairly divide the stakes of an interrupted game of dice based on the current score and the mathematical likelihood of each player winning had the game continued. Pascal initiated a series of letters with his contemporary, Pierre de Fermat. Together, through their correspondence regarding this specific dice problem, Pascal and Fermat developed the fundamental laws of expected value and combinatorial probability. Their work on this dice problem laid the absolute foundation for modern probability theory, statistics, and eventually the fields of quantum mechanics, thermodynamics, and modern economics.

Key Concepts and Terminology in Dice Probability

To accurately calculate and discuss dice probabilities, one must first master the specific terminology and foundational concepts used in statistics and discrete mathematics. The most fundamental concept is the "Sample Space," which represents the complete, exhaustive list of every single possible outcome that can occur in a given scenario. For a single standard six-sided die (abbreviated as 1d6), the sample space is {1, 2, 3, 4, 5, 6}. An "Event" is any specific outcome or collection of outcomes within that sample space that you are trying to measure. For example, rolling an even number on a d6 is an event consisting of the outcomes {2, 4, 6}. The "Probability" of an event is a mathematical value between 0 and 1 (or 0% and 100%) that expresses the likelihood of that event occurring. A probability of 0 means the event is absolutely impossible, while a probability of 1 means the event is absolutely mathematically certain.

Another critical concept is "Independent Events," which describes a scenario where the outcome of one random event has absolutely no effect on the outcome of a subsequent random event. Dice rolls are perfectly independent; rolling a 6 on your first attempt does not change the mathematical probability of rolling a 6 on your second attempt. "Mutually Exclusive Events" refer to two outcomes that cannot possibly occur at the exact same time. On a single roll of a d20, rolling a natural 20 and rolling a natural 1 are mutually exclusive; you cannot do both simultaneously.

When analyzing the long-term behavior of dice, mathematicians rely on "Expected Value" (EV), which is the mathematically predicted average result if you were to roll the dice an infinite number of times. Finally, "Variance" and "Standard Deviation" measure the statistical spread or dispersion of the outcomes. A system with low variance will consistently produce results very close to the expected value, while a system with high variance will wildly swing between the absolute minimum and maximum possible outcomes. Understanding these terms is non-negotiable for anyone attempting to map out the complex statistical reality of dice mechanics.

How It Works — Step by Step

Calculating dice probability relies on combinatorial mathematics, which is the study of counting, arranging, and combining sets of elements. For a single die, the fundamental formula is incredibly straightforward: the probability of an event $P(E)$ equals the number of target outcomes divided by the total number of possible outcomes in the sample space. If you want to roll a 5 or higher on a 20-sided die (d20), your target outcomes are the numbers 5 through 20 (which is 16 distinct numbers). The total possible outcomes are 20. Therefore, $P(E) = 16 / 20 = 0.80$, or an exactly 80% chance. This basic formula works perfectly for any single die, provided the die is perfectly balanced and every face has an equal chance of appearing.

However, when you begin adding multiple dice together, the mathematics becomes significantly more complex because the distribution of outcomes changes from a flat line to a bell curve. To calculate the exact probability of rolling a specific sum ($k$) using a specific number of dice ($n$) that each have a specific number of sides ($s$), mathematicians use a generating function or a specialized combinatorial formula. The exact mathematical formula is: $P(k, n, s) = \frac{1}{s^n} \sum_{i=0}^{\lfloor (k-n)/s \rfloor} (-1)^i \binom{n}{i} \binom{k - si - 1}{n - 1}$ In this formula, $s^n$ represents the total possible combinations. The symbol $\Sigma$ represents a summation of the sequence. The $\lfloor \rfloor$ symbols indicate the floor function (rounding down to the nearest whole number). The $\binom{n}{i}$ represents a binomial coefficient, calculated as $n! / (i!(n-i)!)$, where the exclamation point denotes a factorial.

Let us walk through a complete, fully worked example using realistic numbers: calculating the exact probability of rolling a sum of exactly 10 using three six-sided dice (3d6). First, identify the variables: $n = 3$ (number of dice), $s = 6$ (sides per die), and $k = 10$ (target sum). Step 1: Calculate the total possible outcomes, which is $s^n$, or $6^3 = 216$. Step 2: Determine the upper limit of our summation: $\lfloor (10-3)/6 \rfloor = \lfloor 7/6 \rfloor = 1$. So we will calculate the summation for $i = 0$ and $i = 1$. Step 3: Calculate the term for $i = 0$. $(-1)^0 \times \binom{3}{0} \times \binom{10 - 6(0) - 1}{3 - 1}$ $= 1 \times 1 \times \binom{9}{2}$. $\binom{9}{2} = 9! / (2! \times 7!) = (9 \times 8) / 2 = 36$. Step 4: Calculate the term for $i = 1$. $(-1)^1 \times \binom{3}{1} \times \binom{10 - 6(1) - 1}{3 - 1}$ $= -1 \times 3 \times \binom{3}{2}$. $\binom{3}{2} = 3! / (2! \times 1!) = 3$. $-1 \times 3 \times 3 = -9$. Step 5: Add the terms together: $36 + (-9) = 27$. There are exactly 27 unique combinations of three six-sided dice that add up to 10. Step 6: Divide by the total outcomes: $27 / 216 = 0.125$. Therefore, the exact mathematical probability of rolling a 10 on 3d6 is 0.125, or exactly 12.5%.

Types, Variations, and Methods

The mathematical approach required to calculate dice probability changes drastically depending on the specific rolling method dictated by the game's rules. The most basic method is the "Single Die Roll," commonly used in games like Dungeons & Dragons (using a 1d20). This produces a "discrete uniform distribution," meaning every single number has the exact same probability of appearing. On a d20, rolling a 1 has a 5% chance, rolling a 10 has a 5% chance, and rolling a 20 has a 5% chance. The probability graph for this method is a perfectly flat horizontal line. This creates highly volatile, swingy gameplay where extreme outcomes are just as likely as average outcomes.

The second major variation is the "Sum of Multiple Dice," such as rolling 2d6 in Monopoly or Settlers of Catan. This method produces a "triangular distribution" (for two dice) or a "normal distribution" (a bell curve, for three or more dice). Because there are many ways to roll a middle number (e.g., a 7 on 2d6 can be 1+6, 2+5, 3+4, 4+3, 5+2, 6+1) but only one way to roll the extremes (e.g., a 2 must be 1+1), the probabilities heavily cluster around the mathematical average. This creates consistent, predictable gameplay where extreme results are incredibly rare.

Dice Pools and Success Counting

Another prevalent method is the "Dice Pool" system, used in games like Shadowrun or Vampire: The Masquerade. Instead of adding the numbers together, the player rolls a large number of dice and counts how many individual dice meet or exceed a specific target number (called "successes"). The probability for this is calculated using the Binomial Distribution formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$, where $n$ is the number of dice rolled, $k$ is the target number of successes, and $p$ is the probability of a single die being a success.

Advantage and Disadvantage Mechanics

Modern game design frequently utilizes "Advantage" or "Disadvantage" mechanics, where a player rolls two dice and keeps the highest (advantage) or lowest (disadvantage). To calculate the probability of rolling at least a target number $x$ on a d20 with advantage, the formula is $P(\text{success}) = 1 - ((x-1)/20)^2$. For example, if you need an 11 or higher, normally you have a 50% chance. With advantage, your chance becomes $1 - (10/20)^2 = 1 - 0.25 = 0.75$, or 75%.

Exploding Dice

Finally, there is the "Exploding Dice" variation. In this system, if a player rolls the maximum possible value on a die, they add that maximum value to their total and get to roll the die again, adding the new result to the running total. If the new roll is also a maximum, it explodes again, theoretically continuing to infinity. Because the die can explode infinitely, calculating the exact expected value requires solving an infinite geometric series. For a die with $s$ sides, the Expected Value of an exploding die is $EV = (\text{Normal } EV) \times (\frac{s}{s-1})$. For a standard d6 (normal EV of 3.5), an exploding d6 has an expected value of $3.5 \times (6/5) = 4.2$.

Real-World Examples and Applications

To truly understand dice probability, one must see how these mathematical formulas apply to concrete, real-world scenarios across various games and industries. Let us examine a classic combat scenario in Dungeons & Dragons 5th Edition. A player character is attempting to strike an enemy with an Armor Class (AC) of 16. The player has a +6 bonus to their attack roll. To determine the probability of a successful hit, we first determine the target number the die must show. The player needs a total of 16, and they already have 6, so the physical 20-sided die must show a 10 or higher. The outcomes 10 through 20 represent 11 winning faces on the die. Therefore, $11 / 20 = 0.55$. The player has an exactly 55% chance to hit the enemy on any given attack.

In the high-stakes environment of a casino, dice probability dictates millions of dollars in daily revenue, specifically at the Craps table. The foundational bet in Craps is the "Pass Line" bet on the "come-out roll" using two six-sided dice. The player instantly wins if they roll a 7 or 11, and instantly loses if they roll a 2, 3, or 12. There are 6 ways to roll a 7, and 2 ways to roll an 11, totaling 8 winning combinations. There is 1 way to roll a 2, 2 ways to roll a 3, and 1 way to roll a 12, totaling 4 losing combinations. Out of the 36 total possible combinations on 2d6, 8 result in an instant win (22.22%), 4 result in an instant loss (11.11%), and the remaining 24 combinations (66.67%) establish a "point," moving the game into a more complex secondary phase. By mapping out the entire probability tree of the game, casinos ensure that the overall house edge on the Pass Line bet is exactly 1.41%, guaranteeing long-term mathematical profit.

In the classic board game Risk, combat is resolved via opposed dice rolls where the attacker rolls up to three d6s and the defender rolls up to two d6s, comparing the highest dice against each other. Calculating the exact probabilities requires mapping out all $6^5 = 7,776$ possible combinations. When an attacker rolls 3 dice against a defender's 2 dice, the rigorous mathematical breakdown shows that the attacker will win both dice comparisons in 2,890 scenarios (37.17%), the defender will win both comparisons in 2,275 scenarios (29.26%), and they will tie (meaning one die goes to the attacker, one to the defender) in 2,611 scenarios (33.58%). This mathematical reality proves that in Risk, large attacking armies have a definitive statistical advantage over entrenched defenders.

Common Mistakes and Misconceptions

When dealing with probability, human intuition is notoriously flawed, leading beginners and even seasoned professionals to make severe mathematical errors. The single most pervasive misconception is the "Gambler's Fallacy," which is the erroneous belief that independent events are somehow influenced by past outcomes. A player who rolls five consecutive 1s on a d20 will often falsely believe they are "due" for a high roll, assuming the universe will self-correct to maintain the average. In reality, dice have no memory. The mathematical probability of rolling a 20 on the sixth roll remains exactly 5%, completely unaffected by the previous five failures. The Law of Large Numbers dictates that averages will smooth out over tens of thousands of rolls, but it makes absolutely no guarantees about short-term, localized sequences.

Another massive mistake is confusing permutations with combinations when rolling multiple dice. A combination is a set of items where order does not matter, while a permutation is a specific ordered sequence. When rolling two six-sided dice, many beginners assume that rolling a 3 and a 4 is a single outcome. Mathematically, it represents two distinct permutations: Die A rolls a 3 and Die B rolls a 4, OR Die A rolls a 4 and Die B rolls a 3. Failing to count these as two separate, distinct events within the 36-item sample space will completely destroy the accuracy of any probability calculation.

Furthermore, beginners frequently misunderstand how to combine probabilities for multiple independent attempts, often simply adding the percentages together. If a player has a 10% chance to succeed on an action, and they are given two attempts, a novice will incorrectly assume they have a 20% total chance of success (10% + 10%). This additive math is categorically wrong and leads to impossible probabilities (e.g., ten attempts would equal a 100% chance, which is mathematically false). The correct method is to calculate the probability of total failure, and subtract that from 1. The chance of failing a 10% probability check is 90% (0.90). The chance of failing twice in a row is $0.90 \times 0.90 = 0.81$. Therefore, the true probability of succeeding at least once in two attempts is $1 - 0.81 = 0.19$, or exactly 19%.

Best Practices and Expert Strategies

Professionals who rely on probability—such as actuary scientists, professional poker players, and lead game designers—do not simply calculate odds; they employ specific mental frameworks and best practices to apply those odds effectively. The most important expert strategy is to always calculate "Cumulative Probability" rather than exact probability. In almost all practical scenarios, you do not care about rolling exactly a specific number; you care about rolling at least that number or higher. Knowing that the exact probability of rolling precisely a 15 on 3d6 is 4.62% is largely useless information. Knowing that the cumulative probability of rolling a 15 or higher on 3d6 is 9.25% is actionable data that dictates whether you should take a risk.

Experts also rigorously utilize Expected Value (EV) as their primary decision-making metric. Expected value cuts through the noise of individual random events and provides a hard mathematical baseline for the true worth of an action. To calculate EV, you multiply every possible outcome by its probability of occurring, and sum the results. For a standard d6, the math is $(1 \times 1/6) + (2 \times 1/6) + (3 \times 1/6) + (4 \times 1/6) + (5 \times 1/6) + (6 \times 1/6) = 3.5$. By comparing the Expected Value of different options, experts can mathematically prove which choice is superior over the long term, regardless of short-term variance.

Additionally, expert practitioners always map out the boundary conditions—the absolute best-case and absolute worst-case scenarios—before engaging with a probabilistic system. They calculate the mathematical floor (minimum possible result) and the ceiling (maximum possible result) to understand the system's variance. A weapon in a game that deals 1d12 damage and a weapon that deals 2d6 damage both have similar maximums (12), but drastically different minimums (1 vs 2) and drastically different expected values (6.5 vs 7). By analyzing these boundaries, experts can choose low-variance options when they need guaranteed, consistent performance, and high-variance options when they are mathematically behind and require a lucky, extreme outcome to succeed.

Edge Cases, Limitations, and Pitfalls

While mathematical formulas provide perfect answers in a theoretical vacuum, real-world probability is fraught with physical and computational edge cases that can invalidate theoretical math. The most significant physical limitation is the reality of imperfect dice. Theoretical probability assumes a perfectly balanced, mathematically ideal polyhedron. In reality, mass-produced plastic dice are tumbled in polishing machines, which unevenly sands down the edges and corners. Furthermore, opaque plastic dice frequently contain internal air bubbles that subtly shift the center of gravity, causing the die to favor the faces opposite the bubble. A standard hobbyist d20 might actually roll a 20 only 3% of the time, and roll a 1 up to 7% of the time, completely skewing any mathematical calculations based on a perfect 5% uniform distribution.

From a computational standpoint, a major pitfall occurs when attempting to calculate exact probabilities for massive dice pools. Calculating the exact probability distribution for rolling 3d6 is trivial, requiring an analysis of 216 combinations. However, if a game requires a player to roll 100 six-sided dice and calculate the probability of scoring exactly a sum of 350, the sample space expands to $6^{100}$ possible combinations—a number so incomprehensibly large that it exceeds the number of atoms in the observable universe. Attempting to use standard combinatorial formulas for this will cause integer overflow errors in standard computing systems.

Another fascinating mathematical edge case involves "Non-Transitive Dice," such as Grime Dice or Efron's Dice. In standard logic, if A beats B, and B beats C, then A must beat C. Non-transitive dice mathematically break this rule. You can design a set of three specific dice where Die A has a mathematical advantage over Die B, Die B has an advantage over Die C, but Die C has an advantage over Die A, creating a rock-paper-scissors loop of probability. Failing to recognize a non-transitive probability structure is a severe pitfall that can lead to completely erroneous assumptions about statistical superiority.

Industry Standards and Benchmarks

Across various industries that rely on dice and probability, specific mathematical benchmarks and physical standards have been established as the definitive norms. In the casino industry, the physical standards for dice are incredibly strict to ensure mathematical theory matches physical reality. Casino dice are manufactured to astonishingly precise tolerances, typically machined to within 0.0001 inches of perfect symmetry. They are made from translucent cellulose acetate so that inspectors can verify there are no internal weights or air bubbles, and the pips (the dots) are drilled out and filled with a material of the exact same density as the surrounding plastic to ensure the die's center of gravity remains perfectly dead-center.

In the tabletop RPG and video game design industries, probability benchmarks focus heavily on human psychology and player satisfaction. Decades of playtesting data have established the "70% Success Rule" as an industry standard for game balance. Psychological studies show that human beings are terrible at intuitively understanding probability; if a player has a 50% chance to succeed, they will psychologically perceive it as failing constantly. Game designers have found that tuning the baseline mathematical probability of a standard, competent action to roughly 65% to 70% success creates the ideal psychological feeling of competence and fairness for the player.

Furthermore, the industry standard for modern RPG design relies heavily on the concept of "Bounded Accuracy," a mathematical framework popularized by Dungeons & Dragons 5th Edition. In this framework, the mathematical floor and ceiling of the game are strictly capped. A starting character might have a +5 bonus to a d20 roll, while a maximum-level, god-like character might only have a +11 bonus. Because the 1d20 provides a massive 1-to-20 variance, the strict mathematical boundaries on the static bonuses ensure that a low-level threat always retains a statistical chance (even if it is just 5%) of hitting a high-level target. This prevents the math from scaling into infinity and keeps the probability calculations manageable and relevant at all stages of the game.

Comparisons with Alternatives

When attempting to determine the likelihood of a complex random event, exact mathematical calculation via combinatorial formulas is not the only approach. The primary alternative is the "Monte Carlo Simulation," a computational algorithm that relies on repeated random sampling to obtain numerical results. Instead of trying to algebraically solve the exact formula for a wildly complex scenario (e.g., rolling 8d6, rerolling all 1s and 2s, but only keeping the highest 5 dice), a Monte Carlo simulation simply uses a computer program to virtually roll those dice 1,000,000 times, records the results, and provides the statistical average.

The choice between Exact Mathematical Calculation and Monte Carlo Simulation comes down to a trade-off between absolute precision and computational feasibility. Exact math is vastly superior when dealing with small numbers of dice or simple sum calculations. It provides a perfect, irrefutable, rational number (like precisely 1/6) without requiring any programming knowledge or computing power. It allows mathematicians to algebraically prove a concept.

However, Exact Math completely breaks down when introducing conditional rerolls, exploding dice, and massive dice pools. A game mechanic that says "Roll 10d10, count successes on 8s or higher, 10s explode, 1s subtract a success" requires an algebraic formula so complex that it would take hours to write out, with a massive margin for human error. In these scenarios, Monte Carlo simulations are infinitely superior. While a simulation with 1,000,000 iterations will only provide an approximation (e.g., 45.231% instead of the exact mathematical 45.231459%), the approximation is statistically perfect for any real-world application, and the computer can generate the answer in milliseconds. Therefore, exact math is the tool of the theorist and the student, while Monte Carlo simulations are the workhorse of the modern software developer and professional game designer.

Frequently Asked Questions

What is the most common roll on 2d6, and exactly how likely is it? The most common mathematical sum when rolling two six-sided dice is 7. Out of the 36 total possible combinations, there are exactly 6 ways to roll a 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). To calculate the probability, you divide the 6 target outcomes by the 36 total outcomes, resulting in 6/36, which simplifies to 1/6. Therefore, you have an exactly 16.67% chance of rolling a 7 on any given roll of 2d6.

How does "Advantage" actually change the mathematical average of a d20 roll? Rolling a standard 1d20 yields a mathematical expected value (average) of exactly 10.5. When rolling with Advantage (rolling two d20s and keeping the highest result), you drastically alter the probability curve, shifting the weight heavily toward the higher numbers. The new expected value of rolling a d20 with advantage is mathematically calculated to be 13.825. Therefore, gaining advantage on a roll is roughly mathematically equivalent to gaining a static +3.325 bonus to your roll.

What does a notation like "3d6" actually mean in statistical terms? In standard dice notation, the first number represents the total quantity of dice being rolled, the "d" stands for die/dice, and the final number represents the number of faces on those dice. Therefore, "3d6" means rolling three independent, six-sided dice and summing their results. Statistically, this notation defines the parameters of the sample space: it indicates a system with $6^3$ (216) total possible permutations, generating a bell-curve distribution with a minimum possible sum of 3, a maximum possible sum of 18, and a mathematical expected value of exactly 10.5.

How do you calculate the probability of rolling at least one specific number, like a 6, on multiple dice? The easiest and most mathematically sound way to calculate "at least one" is to calculate the probability of it never happening, and subtracting that from 100%. If you roll three six-sided dice, the chance of a single die NOT being a 6 is 5/6. The chance of all three dice not being a 6 is $(5/6) \times (5/6) \times (5/6) = 125 / 216$, which is roughly 57.87%. By subtracting this failure rate from 1 (or 100%), we find that $1 - 0.5787 = 0.4213$. You have an exactly 42.13% chance of rolling at least one 6 when rolling 3d6.

Is rolling a natural 1 on a d20 really a 5% chance? It feels like it happens more often. Mathematically, on a perfectly balanced, mathematically ideal 20-sided die, rolling a 1 is exactly a 1/20 chance, or 5.00%. However, human perception is heavily biased by the "negativity bias," a psychological phenomenon where humans remember negative events much more vividly than mundane ones. Additionally, physically manufactured dice are almost never perfectly balanced; tumbling processes and internal air bubbles can easily skew a cheap plastic d20 to roll a 1 as much as 7% or 8% of the time.

What are exploding dice, and how do they affect the mathematical average? Exploding dice is a probability mechanic where rolling the maximum value on a die allows you to add that value to your total and immediately roll the die again, adding the new result. Because the die can theoretically explode infinitely, it raises the overall mathematical average of the die. The formula for the new expected value is the standard average multiplied by $(s / (s-1))$, where $s$ is the number of sides. For an 8-sided die (normal average 4.5), an exploding mechanic raises the true mathematical average to $4.5 \times (8/7)$, or approximately 5.14.