Calculate single-event probability, combine two independent events, or find a complement probability — instantly.
Enter favorable and total outcomes to see the probability
Enter P(A) and P(B) to combine two independent events
Enter P(A) to find its complement
This free probability calculator covers the three most common probability questions in one tool: finding the probability of a single event from favorable and total outcomes, combining two independent events with the AND/OR rules, and finding a complement probability. Whether you need a quick favorable outcomes calculator for a homework problem, an independent events probability calculator for a statistics course, or a complement probability calculator for a risk model, this tool shows the exact formula and the arithmetic behind every result, displayed as a decimal, a simplified fraction, and a percentage.
Probability quantifies how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). The Single Event tab measures the probability of an event as favorable outcomes divided by total outcomes. The Two Independent Events tab measures how two separate, unrelated events combine — either both happening (AND) or at least one happening (OR). The Complement tab measures the probability that an event does not happen.
This tool is built for students working through probability of an event homework, teachers building practice problems, data scientists sanity-checking a quick probability estimate, risk and insurance analysts estimating exposure, quality-control engineers computing defect rates, and anyone curious about the odds behind a game, a coin flip, or a dice roll. No statistics background is required — just the numbers for your specific question.
Probability is the mathematical language of uncertainty, and it underlies decision-making in nearly every quantitative field. Insurers price policies based on the probability of a claim. Manufacturers use probability to model defect rates. Meteorologists combine probabilities of independent atmospheric conditions to forecast weather. Understanding whether events are independent, and correctly applying the AND/OR/complement rules, is often the difference between a correct and an incorrect probability estimate.
Casinos and game designers use probability to set house edges and payout odds. Insurance actuaries use complement probability to compute the likelihood of "at least one" claim across a policy pool. Quality-control teams multiply independent defect probabilities across production stages. Geneticists use independent-event multiplication to compute the odds of inheriting multiple traits. Even everyday decisions — like whether to carry an umbrella — rely on probability estimates.
How this probability calculator turns your inputs into a probability, a joint/union probability, or a complement
Every valid probability sits between 0 (impossible) and 1 (certain). Multiply by 100 to express it as a percentage.
The AND/OR formulas on this page only hold when A and B are independent — verify this before applying them.
P(A) + P(not A) = 1 always. This shortcut is often faster than computing an event's probability directly.
From picking a mode to verifying your result
Choose Single Event, Two Independent Events, or Complement, depending on the question you're solving.
For Single Event, enter favorable and total outcomes. For Two Independent Events, enter P(A) and P(B) as decimals. For Complement, enter just P(A).
The calculator applies the matching probability formula instantly and validates that your inputs are within a valid range (0–1, or favorable ≤ total).
Single Event shows a decimal, a simplified fraction, and a percentage. Two Independent Events shows P(A and B) and P(A or B). Complement shows P(not A).
Multi-step problems often need more than one tab — for example, find a single-event probability first, then feed it into Two Independent Events or Complement.
Recompute using the complement rule (1 minus the result) or the multiplication rule to confirm the calculator's output matches your own working.
Rolling an even number on a die, combined with flipping heads on a coin
A fair six-sided die is rolled once and a fair coin is flipped once. Event A = "the die shows an even number" (2, 4, or 6). Event B = "the coin lands heads." What is P(A), P(not A), P(A and B), and P(A or B)?
Explanation: Because the die roll and coin flip are independent, the AND probability (0.25) is simply the product of the two individual probabilities — smaller than either input, as it always should be for two events that both need to happen. The OR probability (0.75) is larger than either input, since it captures "at least one" of the two events occurring, and the subtraction of 0.25 prevents the overlap (both A and B happening) from being counted twice.
What each mode's output actually represents
| Mode | What the Result Means | Example Reading |
|---|---|---|
| Single Event | The chance the event occurs, as a decimal, simplified fraction, and percentage | 3/6 = 0.5 = 50% chance of rolling even |
| P(A and B) | The chance both independent events occur together — always ≤ the smaller of P(A), P(B) | 0.5 × 0.5 = 0.25 → 25% chance of both |
| P(A or B) | The chance at least one of the two independent events occurs — always ≥ the larger of P(A), P(B) | 0.5 + 0.5 − 0.25 = 0.75 → 75% chance of at least one |
| P(not A) | The chance the event does not occur — always equals 1 minus P(A) | 1 − 0.35 = 0.65 → 65% chance A does not happen |
Typical ranges: a probability near 0 means the event is rare or unlikely; near 1 means it's nearly certain; exactly 0.5 means it's a coin-flip, equally likely to happen or not. P(A and B) for independent events is always less than or equal to either individual probability, while P(A or B) is always greater than or equal to either individual probability.
Manual verification: for the Single Event tab, multiply your simplified fraction back out and confirm it matches the original favorable/total ratio. For AND/OR, check that P(A and B) ≤ min(P(A), P(B)) ≤ max(P(A), P(B)) ≤ P(A or B) ≤ 1 — this ordering must always hold for valid independent-event probabilities.
Where a quick, reliable probability calculation is genuinely useful
Calculate the odds of winning a game, drawing a specific card, or rolling a dice combination.
Estimate the likelihood of a claim or event to help price policies and assess exposure.
Compute the probability of a defect occurring at one or multiple stages of production.
Combine probabilities of independent atmospheric conditions into a combined forecast likelihood.
Sanity-check probability estimates used in classifiers, A/B tests, and simple event models.
Estimate the probability of independent market events or risk scenarios occurring together.
Check homework on the probability of an event, independent events, and complements.
Practice probability word problems commonly found on entrance and placement exams.
Estimate the odds of independently inherited traits appearing together in offspring.
Work out the odds behind a specific draw, roll, or combination during gameplay.
Estimate the combined probability of independent security failures or breach events.
Compute failure probabilities across independent components in a system or process.
Estimate the probability of independent test results or screening outcomes occurring together.
What this probability calculator does well, and where manual judgment is still needed
Three related concepts that are frequently confused
| Concept | Definition | Formula |
|---|---|---|
| Independent Events | The occurrence of one event doesn't affect the probability of the other | P(A and B) = P(A) × P(B) |
| Mutually Exclusive Events | The two events cannot both happen at the same time | P(A and B) = 0, so P(A or B) = P(A) + P(B) |
| Conditional Probability | The probability of A occurring given that B has already occurred | P(A|B) = P(A and B) ÷ P(B) |
Summary: This probability calculator gives you an instant, free way to find single-event probability, combine two independent events with AND/OR, and compute a complement — all with the formula and simplified fraction shown. Pair it with related tools like the Z-Score Calculator and Normal Distribution Calculator when you're working with continuous, normally-distributed data instead of discrete favorable/total counts.
Common questions about probability
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