Probability Calculator
Single events, A and B, A or B, and the chance of at least once
๐ฒ What do you want to find?
๐ฏ Single event
๐ Result
๐งฎ Breakdown
All results assume the outcomes are equally likely (for a single event) and that events are independent (for the two-event and repeated-trial modes). For exact counts of successes use a binomial calculator.
Last updated June 2026
Method: Standard probability rules - single-event P = favorable / total, the multiplication rule P(A and B) = P(A) x P(B), the addition rule P(A or B) = P(A) + P(B) - P(A) x P(B), and the complement rule P(at least once) = 1 - (1 - p)^n. Results are shown as a reduced fraction, a decimal and a percent.
Included: Single events, two independent events (and, or, exactly one, neither), and the chance an event happens at least once over n trials, with a worked breakdown for each.
Not included: Dependent events (no replacement / conditional probability), the binomial probability of exactly k successes, and continuous distributions. These need their own tools.
Probability calculator: everything you need to know
If you roll a fair six-sided die, the probability of getting a 4 is exactly 1 in 6 - which is 0.1667 as a decimal and about 16.67% as a percent. That single calculation, favorable outcomes divided by total outcomes, is the foundation of all probability. This probability calculator takes that idea and extends it to the questions people actually ask: what are the odds of two things both happening, of either one happening, and of something rare happening at least once if you try enough times? It returns every answer as a fraction, a decimal, and a percent so the result is easy to read no matter the context.
The core formulas
Probability is a number between 0 and 1. A 0 means the event is impossible; a 1 means it is certain. The four rules this tool uses are:
Single event: P = favorable ÷ total Both (and): P(A and B) = P(A) × P(B) Either (or): P(A or B) = P(A) + P(B) − P(A) × P(B) At least once: P = 1 − (1 − p)n The multiplication and addition rules above assume the events are independent - one does not change the other. The "at least once" rule uses the complement: instead of adding every way the event could happen, it works out the single probability that it never happens and subtracts that from 1.
How to use this probability calculator
Pick the mode that matches your question, then fill in the inputs:
- Choose a mode: "Single event," "Two events," or "Repeated trials" at the top.
- Single event: enter how many outcomes count as a success (favorable) and how many outcomes are possible in total. The quick buttons load common setups like a coin or a die.
- Two events: enter P(A) and P(B) as decimals between 0 and 1. The calculator shows the percent next to each field so you can double-check.
- Repeated trials: enter the probability per try and how many tries (n), and read the chance it happens at least once.
The result updates instantly. The large number at the top is the headline answer, and the cards beneath it give the fraction, decimal, and percent plus a full breakdown of the related probabilities.
Worked example 1: a single die roll
What is the probability of rolling an even number on a die? The even faces are 2, 4, and 6 - that is 3 favorable outcomes out of 6. So P = 3/6, which reduces to 1/2 = 0.5 = 50%. The complement (rolling an odd number) is also 50%, because the two possibilities cover every outcome and must add to 1.
Worked example 2: two coins (and / or)
Flip two fair coins. Each has a 0.5 chance of heads. The probability that both land heads is P(A and B) = 0.5 × 0.5 = 0.25 (25%). The probability that at least one is heads is P(A or B) = 0.5 + 0.5 - 0.25 = 0.75 (75%). Notice we subtracted the 0.25 overlap - without that subtraction you would wrongly get 100%, which would mean heads is guaranteed. It is not: there is a 25% chance both coins come up tails.
Worked example 3: at least once over n tries
Suppose a game has a 10% (p = 0.1) drop chance each time you play. Over 10 plays, your intuition might say 100% - but that is wrong. The chance of never getting the drop is 0.910 = 0.349, so the chance of getting it at least once is 1 - 0.349 = 0.651, about 65%. You would actually need around 22 plays to cross 90%, and rare events never truly reach 100% no matter how many tries you make.
Quick reference table
Common single-event probabilities, shown three ways:
| Event | Fraction | Decimal | Percent |
|---|---|---|---|
| Coin lands heads | 1/2 | 0.5 | 50% |
| Roll a specific number on a die | 1/6 | 0.1667 | 16.67% |
| Roll an even number | 1/2 | 0.5 | 50% |
| Draw an ace from a deck | 1/13 | 0.0769 | 7.69% |
| Draw a specific card | 1/52 | 0.0192 | 1.92% |
| Both of two coins are heads | 1/4 | 0.25 | 25% |
| At least one head in two flips | 3/4 | 0.75 | 75% |
Who this calculator is for
- Students checking homework on basic probability, the and/or rules, and the complement.
- Gamers and hobbyists working out drop rates, loot odds, or the chance of a result over many attempts.
- Anyone making decisions under uncertainty - estimating the chance that at least one of several things goes wrong (or right).
- Teachers who want a clean way to show the same probability as a fraction, decimal, and percent.
Key probability terms
- Outcome: a single possible result, such as rolling a 4.
- Sample space: the full set of possible outcomes, such as {1, 2, 3, 4, 5, 6} for a die.
- Favorable outcome: an outcome that counts as a success for the event you are measuring.
- Independent events: events where one result has no effect on the other.
- Complement: the event "this does not happen," with probability 1 - P.
- Mutually exclusive: events that cannot both occur at once; for those, P(A or B) = P(A) + P(B) with nothing to subtract.
Independent vs. dependent events
The multiply-and-add formulas in this tool assume independence. That holds for coins, dice, and spinners. It breaks down when one event changes the next - for example, drawing two cards without replacing the first. There, the probability of the second draw depends on what you drew first, so you need conditional probability, P(B given A), rather than the simple product. If your events are linked, treat this calculator's two-event mode as an approximation only.
Why "at least once" surprises people
People routinely overestimate how fast a rare event becomes near-certain. A 1% chance per attempt does not reach 50% until about 69 attempts, and never quite hits 100%. The complement rule, 1 - (1 - p)n, is the cleanest way to see this. Whenever a question contains the phrase "at least one," reach for the complement rather than trying to add up every individual case - it is faster and far less error-prone.
Tips for getting it right
- Always sanity-check that your answer sits between 0% and 100%.
- For "and" think "smaller" - combining requirements makes an event rarer, so P(A and B) is never larger than either piece.
- For "or" think "bigger" - P(A or B) is at least as large as the larger of the two.
- Convert percentages to decimals (divide by 100) before multiplying.
- Use the complement for any "at least one" question.
Related concepts and tools
Probability sits next to several other math topics. For exactly k successes in n trials (like "exactly 3 heads in 5 flips"), you need the binomial distribution, which builds on combinations. To turn a probability into a percentage of a quantity, the Percentage Calculator helps; to simplify the fraction form of a result, the Fraction Calculator does the reducing; and powers like (1 - p)n are exactly what the Exponent Calculator evaluates. This page focuses on the everyday rules - single, combined, and "at least once" - that cover the large majority of probability questions.
โ ๏ธ Common mistakes & edge cases
Adding probabilities without subtracting the overlap
For "A or B," people often add P(A) + P(B) and stop. With two coins that gives 0.5 + 0.5 = 1.0, implying heads is guaranteed - it is not. You must subtract P(A and B): 0.5 + 0.5 - 0.25 = 0.75.
Treating dependent events as independent
Drawing two cards without putting the first back is not independent - the second draw's odds change. Multiplying the original probabilities gives the wrong answer; you need conditional probability instead.
Assuming "at least once" reaches 100%
A 10% chance over 10 tries is about 65%, not 100%. Repeated trials raise the odds but never guarantee a rare event. Use 1 - (1 - p)^n rather than n × p, which can even exceed 1.
Mixing up percent and decimal
Entering 25 instead of 0.25 inflates every result by 100x and pushes it past 1. Convert percentages to decimals first; the calculator flags any value outside the valid 0-to-1 range.
❓ Frequently asked questions
How do you calculate the probability of a single event?
Divide the number of favorable outcomes by the total number of equally likely outcomes: P = favorable / total. For example, rolling a 4 on a fair six-sided die has 1 favorable outcome out of 6, so P = 1/6 = 0.1667 = 16.67%. Probability is always between 0 (impossible) and 1 (certain).
What is the probability of two events both happening?
For two independent events, multiply their probabilities: P(A and B) = P(A) x P(B). If P(A) = 0.5 and P(B) = 0.5, then P(A and B) = 0.5 x 0.5 = 0.25, or 25%. 'Independent' means one event does not affect the other - like flipping two separate coins.
How do you calculate the probability of A or B?
For independent events, P(A or B) = P(A) + P(B) - P(A) x P(B). You subtract the overlap so you do not count the case where both happen twice. With P(A) = 0.5 and P(B) = 0.5, P(A or B) = 0.5 + 0.5 - 0.25 = 0.75, or 75%.
What is the 'at least once' probability over many trials?
The easiest route is the complement. The chance the event never happens in n independent trials is (1 - p)^n, so the chance it happens at least once is 1 - (1 - p)^n. With p = 0.1 over 10 trials, that is 1 - 0.9^10 = 1 - 0.349 = 0.651, or about 65%.
What does 'independent events' actually mean?
Two events are independent when the outcome of one does not change the probability of the other. Flipping a coin twice, or rolling two dice, gives independent events. Drawing two cards without replacing the first does not - the second draw depends on the first, so the simple multiply-and-add formulas no longer apply.
What is the difference between probability and odds?
Probability is favorable outcomes over total outcomes (e.g. 1/6). Odds compare favorable to unfavorable outcomes (e.g. 1 to 5, written 1:5). To convert, odds in favor of A are P(A) : (1 - P(A)). A probability of 0.25 is the same as odds of 1:3.
Can a probability be greater than 1 or negative?
No. A valid probability is always between 0 and 1 inclusive (0% to 100%). If a calculation gives a number outside that range, you have either mixed up percent and decimal values or added overlapping events without subtracting their intersection. This calculator flags inputs that fall outside the valid range.
How do I enter a percentage into the calculator?
Convert it to a decimal first by dividing by 100. So 25% becomes 0.25, 5% becomes 0.05, and 100% becomes 1. The calculator shows the percent equivalent next to each field so you can confirm you entered it correctly.
Why use the complement rule for 'at least once'?
Adding up the probability of exactly one success, exactly two, and so on is tedious. The opposite of 'at least once' is simply 'never,' which has one clean value: (1 - p)^n. Subtracting that from 1 gives the answer in a single step, which is why 1 - (1 - p)^n is the standard formula.
Does this calculator handle exactly k successes out of n?
No. This tool covers single events, two independent events, and the 'at least once' case over repeated trials. For the probability of exactly k successes in n trials (the binomial distribution), you need a dedicated binomial calculator, which uses combinations: C(n, k) x p^k x (1 - p)^(n - k).
๐ก Good to know
Probability always lives between 0 and 1
0 means impossible, 1 means certain, and everything in between is a chance. If a result lands outside this range, an input or a formula is off - usually a percent entered as a whole number.
"And" shrinks, "or" grows
Requiring two things to both happen (and) makes the outcome rarer than either alone. Accepting either of two things (or) makes it more likely than either alone. Use this as a gut check on every answer.
The complement is your shortcut
Any "at least one" question is easiest as 1 minus the chance of "none." That single step replaces a long sum of every possible success count, especially over many trials.
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