Probability Calculator

What this probability calculator helps you compare

Probability questions often look similar on the surface but use different rules underneath. Sometimes you are counting favorable outcomes from a clearly defined sample space. Sometimes you want the complement because it is easier to describe what does not happen. In other situations you need the overlap between two events, or you need to update the answer after learning that one event has already occurred.

This page brings those workflows together in one place. The single-event and complement modes are useful when outcomes are equally likely, such as cards, dice, or raffle tickets. The both-events, at-least-one, and conditional modes are more useful when you already know or can estimate event probabilities and need the correct rule for intersections, unions, or event dependence.

How the probability formulas change by question type

The classical rule for a single event is straightforward: divide favorable outcomes by total outcomes. The complement rule then says the chance of not getting that event is 1 minus the event probability. Those two formulas cover a large share of classroom and everyday probability problems because many questions are easiest to solve by counting outcomes or by subtracting from certainty.

Two-event questions are where users often make mistakes. If you want the chance that both events occur and the events are independent, multiply their probabilities. If you want the chance that at least one occurs, add the separate probabilities and subtract the overlap once. If one event changes the meaning of the other, use conditional probability instead: divide the overlap by the event you already know has occurred. This is why P(A|B) and P(B|A) are usually different even when they involve the same two events.

Counting outcomes before calculating probability

Many probability problems start with a counting question: how many outcomes are possible before you can divide favorable outcomes by total outcomes? That is why this probability calculator now includes a combinations and permutations section. It can work as an nCr calculator, an nPr calculator, and a factorial calculator for distinct items chosen without repetition.

Use combinations when order does not matter. A 5-card poker hand is counted with 52 choose 5 because the hand is the same no matter which order the cards were dealt. Use permutations when order matters. First, second, and third place in a 10-runner race are counted with 10P3 because each ordering creates a different result.

Worked examples: die rolls, overlapping events, and conditional probability

Suppose you want the probability of rolling a 6 on a fair die. There is 1 favorable outcome out of 6, so the single-event probability is 1/6, or about 16.67%. The complement, not rolling a 6, is 5/6, or about 83.33%. This is the cleanest kind of probability problem because the outcomes are equally likely and easy to count.

Now suppose a survey says 36% of people own a dog, 30% own a cat, and 14% own both. The probability of owning a dog or a cat is not 36% + 30% = 66%, because the overlap would be double-counted. The correct union probability is 36% + 30% - 14% = 52%. That result is useful because it also lets you break the situation into only-dog, only-cat, both, and neither segments.

For a conditional example, imagine drawing one card from a standard deck and asking for the probability that it is red given that it is a face card. There are 12 face cards total and 6 of them are red, so P(red|face card) = 6/12 = 0.5. That does not mean P(face card|red) is also 0.5. Among the 26 red cards, only 6 are face cards, so P(face card|red) = 6/26, or about 23.08%. The direction of the condition changes the denominator and often changes the answer materially.

Worked examples: 52 choose 5 and 10P3

For combinations, the classic example is 52 choose 5. A poker hand is an unordered group, so C(52,5) = 2,598,960. If you counted the same five cards in every possible order, you would be answering a different question and the count would inflate to P(52,5) = 311,875,200.

For permutations, imagine 10 runners competing for first, second, and third place. The order matters, so P(10,3) = 10 x 9 x 8 = 720. If the same 10 people were being chosen for a 3-person committee, the order would not matter and C(10,3) = 120. The factorial term 3! = 6 explains the difference: each committee of three can be ordered six ways.

How to interpret probability, percentages, odds, and one-in language

Probability itself is usually written as a decimal between 0 and 1. Many users prefer a percentage because it is easier to compare quickly, and some prefer a one-in form because it makes rare events easier to visualize. A probability of 0.2 can be described as 20% or roughly 1 in 5. The underlying chance is the same; only the presentation changes.

Odds are related to probability but are not identical. Probability compares favorable outcomes with all outcomes. Odds compare favorable outcomes with unfavorable outcomes. That is why a probability of 1/6 corresponds to odds of 1 to 5, not 1 to 6. This page reports both the probability and supporting interpretations so you can choose the format that makes the result easiest to explain or compare.

When to use a distribution calculator instead

This page covers event probability rules and counting setup. It does not replace a binomial distribution calculator, normal distribution calculator, Poisson distribution calculator, or broader probability distribution calculator. Those pages answer model-specific questions such as the probability of exactly 4 successes in 10 trials, a normal value falling below a cutoff, or a count process producing a given number of events.

The practical split is simple: use this page when you are choosing the right event rule or counting the sample space. Use a distribution calculator when the problem already names a distribution, has repeated trials, includes a mean and standard deviation, or asks for a cumulative probability under a statistical model.