Probability Calculator

A probability calculator computes the likelihood of events occurring. Our probability calculator supports single events, independent events, dependent events, union (A or B), intersection (A and B), and conditional probability with detailed step-by-step explanations.

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. For multiple events, use the addition rule (P(A or B)) or multiplication rule (P(A and B)) depending on whether events are mutually exclusive or independent.

Events are independent if the occurrence of one does not affect the likelihood of the other occurring. For example, flipping a coin and rolling a die are independent because the coin result does not change the die probability.

Independent events do not influence each other — the probability remains constant. Dependent events change probability based on prior outcomes, such as drawing cards without replacement from a deck.

Conditional probability, written as P(A|B), measures the probability of event A occurring given that event B has already occurred. It is calculated as P(A|B) = P(A ∩ B) / P(B).

A probability of 0 represents absolute impossibility, while a probability of 1 represents absolute certainty. It is impossible to have less than a 0% chance or more than a 100% chance of an event.

The addition rule states P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events (A and B cannot both occur), this simplifies to P(A or B) = P(A) + P(B).

For independent events, P(A and B) = P(A) × P(B). For dependent events, P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given A has occurred.

Yes, this free probability calculator is 100% free with no signup required. Calculate probability for single and multiple events online with step-by-step results instantly.

A standard die has 6 faces, so P(rolling a 6) = 1/6 ≈ 0.1667 or 16.67%. Our probability calculator computes this and much more complex scenarios instantly.

A standard deck has 52 cards with 13 hearts, so P(heart) = 13/52 = 1/4 = 0.25 or 25%.

Mutually exclusive (or disjoint) events cannot occur at the same time. For example, getting heads and tails on a single coin flip are mutually exclusive. Their intersection probability is 0.

Odds in favor = P(event) / P(not event). Odds against = P(not event) / P(event). For example, if P(win) = 0.4, odds in favor = 0.4/0.6 = 2:3.

The complement rule states P(not A) = 1 − P(A). This is useful when calculating the probability that an event does NOT occur, which is often easier than calculating when it does.

You can input probabilities as decimals (0 to 1) or as percentages (0% to 100%). The calculator automatically converts between formats and displays results in both.

The birthday paradox shows that in a group of just 23 people, there is a 50%+ chance that two share a birthday. This counterintuitive result demonstrates how probability can defy intuition.

Bayes' theorem calculates conditional probability: P(A|B) = P(B|A) × P(A) / P(B). It updates probability estimates based on new evidence and is fundamental to medical testing and machine learning.

There are 6 combinations out of 36 total outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). P(sum = 7) = 6/36 = 1/6 ≈ 16.67%.

With replacement, each trial is independent because the total number of outcomes stays constant. Multiply the individual probabilities: P(A and B) = P(A) × P(B).

Without replacement, events are dependent because the total changes. P(A and B) = P(A) × P(B|A), where P(B|A) accounts for the reduced pool.

Since each flip is independent with P(heads) = 0.5, P(3 heads) = 0.5 × 0.5 × 0.5 = 0.125 or 12.5%.

Without replacement: P(2 aces) = (4/52) × (3/51) = 12/2652 ≈ 0.00452 or about 0.45%.

Probability is the ratio of favorable outcomes to total outcomes. Odds compare favorable to unfavorable outcomes. Probability ranges from 0 to 1; odds can range from 0 to infinity.

No. Probability is always between 0 (impossible) and 1 (certain). A value greater than 1 indicates a calculation error.

A probability distribution lists all possible outcomes and their associated probabilities. Common types include binomial, normal (Gaussian), Poisson, and uniform distributions.

The calculator applies the appropriate rule based on event type: multiplication rule for independent events, addition rule for unions, and Bayes' formula for conditional scenarios.

The Probability Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Probability Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.