Probability Calculator

Probability measures how likely an event is on a scale from impossible to certain. A probability of 0 means the event cannot occur under the model, 1 means it must occur, and values between them describe intermediate likelihood. This calculator is designed to handle the most common introductory probability structures without forcing every problem into the same formula.

How to use the Probability Calculator

Start by choosing the probability type that matches the information in your problem. If all outcomes are equally likely and you know how many are favorable, use Favorable outcomes ÷ total outcomes. If you already know P(A) and need the probability that A does not occur, choose the complement mode.

For two-event problems, do not assume independence automatically. Choose Two independent events only when the occurrence of one event does not change the probability of the other. If you know P(A), P(B), and their overlap P(A ∩ B), use the general two-event mode. For a probability “given” another event, choose conditional probability.

Probability inputs accept three convenient formats: decimals such as 0.375, percentages such as 37.5%, and fractions such as 3/8. The calculator converts these to the same internal probability scale and validates that values remain between 0 and 1.

Probability basics and notation

An experiment is a process with uncertain outcomes, a sample space is the set of possible outcomes, and an event is a collection of outcomes of interest. If A is an event, its probability is written P(A).

The sample space S represents all possible outcomes in the model, so its probability is 1. Probabilities can also be written as fractions or percentages: 1/4, 0.25, and 25% are equivalent probability values.

Probability is always tied to a model. A mathematically correct formula can still be a poor real-world model if the assumed outcomes are not equally likely or the event probabilities are estimated badly. The calculator checks arithmetic consistency, but it cannot verify whether your assumptions describe reality.

Probability from favorable and total equally likely outcomes

When a finite sample space has outcomes that are genuinely equally likely, the classical formula is:

If a fair six-sided die is rolled and A means “roll an even number,” the favorable outcomes are {2, 4, 6}. There are 3 favorable outcomes among 6 equally likely outcomes, so P(A) = 3/6 = 1/2.

The phrase equally likely matters. If outcomes have different probabilities, counting outcomes alone is not enough. For example, a weighted spinner can have four labeled regions without each region having probability 1/4.

The complement rule

The complement of A, often written Aᶜ or A′, contains every sample-space outcome that is not in A. Because A and its complement together cover the whole sample space and cannot occur at the same time:

The complement rule is especially useful for “at least one” questions. It can be easier to find the probability of zero successes and subtract from 1 than to add the probabilities of one, two, three, and more successes separately.

For example, if an event happens with probability 0.72, its complement has probability 1 − 0.72 = 0.28. In percentage form, 72% and 28% sum to 100%.

Independent events and the multiplication rule

Events A and B are independent when learning that one occurred does not change the probability of the other. In that special case, the probability that both occur is:

If P(A) = 0.6 and P(B) = 0.4 and the events are independent, P(A and B) = 0.24. Their probability of at least one occurring is 0.6 + 0.4 − 0.24 = 0.76.

Independence should come from the structure of the situation or from justified data, not from convenience. Repeated draws with replacement can often be modeled as independent; repeated draws without replacement are generally dependent because the composition changes after the first draw.

The addition rule for A OR B

For any two events, the probability that A occurs, B occurs, or both occur is:

Why subtract the intersection? Adding P(A) and P(B) counts outcomes in both events twice. Subtracting P(A ∩ B) once corrects that double count.

If A and B are mutually exclusive, their intersection is zero, so the rule simplifies to P(A ∪ B) = P(A) + P(B). But that simplified rule must not be used for overlapping events.

Conditional probability

Conditional probability asks for the probability of A after restricting attention to cases where B is known to have occurred. It is written P(A | B), read “the probability of A given B.”

If 18% of a population satisfies both A and B, while 30% satisfies B, then among the B group the fraction that also satisfies A is 0.18/0.30 = 0.60. So P(A | B) = 60%.

The formula can be rearranged into the general multiplication rule P(A ∩ B) = P(B)P(A | B). Independence is the special situation in which P(A | B) = P(A).

Mutually exclusive is not the same as independent

These terms are frequently confused. Mutually exclusive events cannot happen together, so P(A ∩ B) = 0. Independent events do not affect each other's probabilities.

If A and B are mutually exclusive and both have positive probability, they cannot be independent. Once you know A occurred, B becomes impossible, so knowing A clearly changes the probability of B. The only way two mutually exclusive events can also satisfy the independence equation is when at least one has probability zero.

A familiar example of mutual exclusivity is one fair die roll: “roll a 2” and “roll a 5” cannot occur on the same roll. By contrast, “first coin toss is heads” and “second coin toss is heads” can occur together and are independent under the usual fair independent-toss model.

How the calculator checks whether an overlap is possible

When P(A), P(B), and P(A ∩ B) are entered separately, not every trio of numbers between 0 and 1 is mathematically possible. The intersection cannot exceed either event:

There is also a lower bound because the union cannot exceed 1:

These are sometimes called the Fréchet bounds for two events. For example, if P(A) = 0.8 and P(B) = 0.7, their overlap must be at least 0.5. An entered overlap of 0.2 would force the union to 1.3, which cannot be a probability. SonoCalculator rejects such combinations instead of displaying impossible results.

Worked probability examples

How to interpret a probability result

A probability is not a promise about one individual trial. A 70% event can fail on the next attempt, and a 5% event can occur. Probability describes uncertainty under a model, not certainty for a single outcome.

In repeated comparable trials, a well-calibrated 70% probability is associated with the event occurring about 70% of the time in the long run, but short runs can vary substantially. Real-world estimates also contain modeling error, sampling uncertainty, and changing conditions that a simple arithmetic calculator does not measure.

When communicating probability, percentages are often intuitive, while decimals are convenient for formulas and fractions can preserve exact values. The calculator shows more than one format where useful so you can choose the representation suited to the problem.

Common probability mistakes

Counting outcomes that are not equally likely. Favorable/total counting is valid only when the counted outcomes have equal probability.

Adding probabilities for overlapping events without subtracting the overlap. This double-counts P(A ∩ B).

Multiplying P(A)P(B) without independence. For dependent events, use the general multiplication rule with a conditional probability.

Confusing mutually exclusive with independent. Positive-probability mutually exclusive events actually give information about each other.

Using P(A ∩ B)/P(B) when P(B) = 0. Conditional probability on a zero-probability conditioning event is not defined by this elementary ratio.

Entering percentages as whole numbers without the percent sign. Enter 25% or 0.25, not 25, because a probability must stay between 0 and 1 internally.

Ignoring impossible overlap values. An intersection must satisfy both upper and lower probability bounds. The calculator checks this automatically.

Frequently asked questions

Can probability be greater than 1?

No. A standard probability lies from 0 through 1 inclusive, equivalent to 0% through 100%.

How do I convert probability to a percentage?

Multiply the decimal probability by 100. For example, 0.375 = 37.5%.

What is the probability of not A?

Use the complement rule: P(not A) = 1 − P(A).

How do I find the probability of A and B?

In general, P(A ∩ B) = P(B)P(A | B). If the events are independent, this simplifies to P(A)P(B).

How do I find the probability of A or B?

Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If the events are mutually exclusive, the intersection is zero.

What does “given B” mean?

It means the sample space has been restricted to outcomes where B occurred. P(A | B) measures the fraction of that restricted group that also lies in A.

Can I enter fractions and percentages?

Yes. Probability fields accept forms such as 3/8, 0.375, and 37.5%.

Does this calculator handle permutations or combinations?

This page focuses on probability rules once event counts or probabilities are known. Counting arrangements and selections is a separate task, especially when order matters or when combinations are needed to count favorable outcomes.