Probability Calculator
Single and combined event probability, binomial distribution, and full dice-roll odds β with results shown as a fraction, decimal, percent and odds every time.
β‘ Quick Tips for Getting Probability Right
Features
Favorable Γ· total outcomes, with one-click presets for coins, dice and standard card draws.
Independent AND, mutually exclusive OR, overlapping OR, and conditional P(A|B) β four distinct rules in one mode.
Exact and cumulative probabilities across repeated trials, plus expected value and standard deviation.
Every possible sum's exact probability for up to 4 dice (d4βd20), shown as a bar chart, not one number.
Every result appears as a simplified fraction, decimal, percentage and odds ratio at once.
Every result shows the exact formula and numbers used, not just the final answer.
One Tool for Four Genuinely Different Questions
"What's the probability" changes shape depending on what you're actually asking. Sometimes it's a plain favorable-over-total question β the odds of drawing an Ace. Sometimes it's two events interacting β the odds it rains AND the game gets cancelled. Sometimes it's a repeated trial β how many free throws out of 10 a given shooter is likely to make. And sometimes it's a full distribution, not a single number β every possible sum from rolling dice, not just one target. Most calculators handle one of these well and treat the rest as an afterthought. This tool gives all four their own dedicated mode, sharing the same clean input pattern and the same four-format output (fraction, decimal, percent, odds) so switching between them never feels like changing tools.
Independent, Mutually Exclusive, Overlapping β Picking the Right Rule
These three words get used loosely, but they lead to three different formulas, and picking the wrong one produces a confidently wrong answer. Independent events don't influence each other β two separate coin flips, or a die roll and an unrelated card draw β so P(Aβ©B) is just P(A)ΓP(B). Mutually exclusive events can never both occur β rolling a 2 and rolling a 5 on the same single die β so P(AβͺB) is P(A)+P(B) with nothing to subtract. Overlapping events can both happen β a King or a Heart from a standard deck, since the King of Hearts satisfies both β which means P(A)+P(B) double-counts that shared outcome unless P(Aβ©B) is subtracted back out. Getting this classification right before picking a mode matters more than any arithmetic that follows it.
Binomial Probability Without the Textbook Notation
The formula P(X=k) = C(n,k)Β·pα΅Β·(1βp)βΏβ»α΅ looks intimidating written out, but the idea behind it is simple: C(n,k) counts how many different orders k successes could land in across n trials, and the rest of the formula weighs each of those orders by how likely it actually is. A basketball player who makes 70% of free throws attempting 10 of them has a specific, calculable probability of making exactly 7 β not just "probably around 7," an exact number the binomial formula produces directly, along with the cumulative odds of making at least or at most that many.
Who Actually Reaches for a Probability Calculator
Statistics students checking homework on combined and conditional probability, where a single sign error in a formula produces a confidently wrong final answer that's hard to spot without recomputing it a different way. Game designers and tabletop players balancing dice mechanics β knowing that 2d6 clusters heavily around 7 versus a flat d20's uniform spread changes how a game feels, not just how it scores. Quality control and manufacturing teams using binomial probability to estimate defect rates across a batch of fixed size, where "how many defective units should we expect in 200" has a precise, not approximate, answer. Anyone evaluating a claim in the news or a bet β a "1 in a million" coincidence claim, or combined odds on a parlay bet β benefits from being able to check independence assumptions rather than trusting a headline's arithmetic at face value. And teachers use the dice distribution chart specifically as a visual aid, since seeing the full bell-shaped spread across all eleven sums communicates the idea of a probability distribution faster than a table of numbers does.
Two Calculations Worth Walking Through
Quiz guessing: a 10-question multiple-choice quiz has 4 options per question, so guessing randomly gives a 25% chance per question. Binomial mode with p=25%, n=10 shows the probability of getting exactly 5 right is about 5.8% β low enough to show that "just guess" is a poor strategy, while the expected value (ΞΌ = nΓp = 2.5) confirms most random guessers land around 2β3 correct, not 5.
Craps intuition: Dice Roller with 2d6 shows the sum of 7 has a 16.7% probability β the highest of any sum β while 2 and 12 each sit at just 2.8%. Seeing all eleven possible sums laid out as a bar chart makes it immediately clear why 7 is the reference point for so many dice game rules, rather than something that has to be memorized as a fact.
Why Browser-Based Beats a Statistics Textbook Table
Old-style binomial and probability tables in the back of a textbook only cover a handful of preset n and p values, forcing interpolation for anything in between. This tool computes the exact answer for whatever numbers you enter, instantly, and shows the formula alongside the result so it doubles as a way to check hand-worked homework rather than just replacing it. Nothing you enter is sent to a server β every calculation runs client-side in JavaScript. For the raw combinatorics that power the binomial coefficient, the Permutation & Combination Calculator computes nCr and nPr directly if that's specifically what you need.
What This Calculator Doesn't Cover
It handles discrete, equally-likely-outcome and binomial scenarios β it does not compute continuous distributions like the normal distribution, hypergeometric probability (drawing without replacement from a finite population, where each card or item removed changes the odds for the next one), or Bayesian probability updates that revise an estimate as new evidence arrives. Card-counting scenarios in particular fall outside binomial mode specifically because each draw changes the deck's composition β the trial probability isn't constant, which is the one assumption binomial probability can't relax. Dice rolls are capped at 4 dice and 20 sides per die, which covers every standard tabletop scenario without the performance cost of enumerating truly enormous outcome spaces. For summarizing a real dataset of results rather than a theoretical probability, the Mean, Median & Mode Calculator is the more appropriate tool.