Basic Probability Calculator
This free probability calculator lets you compute single-event probability, the complement rule, the addition rule for unions, and the multiplication rule for independent events — all with instant results.
Quick Formula Reference
- BasicP(A) = favorable / total
- ComplementP(A') = 1 − P(A)
- Union (exclusive)P(A ∪ B) = P(A) + P(B)
- Intersection (independent)P(A ∩ B) = P(A) × P(B)
Quick answer
Probability measures how likely an event is to occur, expressed as a number between 0 and 1 (or 0% to 100%). To find basic probability, divide the number of favorable outcomes by the total number of possible outcomes. The complement of an event A is 1 minus P(A), giving the chance the event does NOT happen. Two independent events multiplied together yield their joint probability, while mutually exclusive events can be added for union probability.
Formula & method
P(A) = favorable outcomes / total outcomes
- P(A) — Probability of event A
- favorable outcomes — Number of outcomes where A occurs
- total outcomes — Total number of equally likely outcomes
Basic single-event probability
P(A') = 1 − P(A)
- P(A') — Probability of the complement (not A)
- P(A) — Probability of event A
Complement rule — probability that A does NOT occur
P(A ∪ B) = P(A) + P(B) [mutually exclusive]
- P(A ∪ B) — Probability that A or B occurs
- P(A) — Probability of event A
- P(B) — Probability of event B
Addition rule for mutually exclusive events (they cannot both occur)
P(A ∩ B) = P(A) × P(B) [independent]
- P(A ∩ B) — Probability that both A and B occur
- P(A) — Probability of event A
- P(B) — Probability of event B
Multiplication rule for independent events (outcome of A does not affect B)
Examples
- Input
- Favorable outcomes = 3 (faces 4, 5, 6); Total outcomes = 6
- Result
- P(A) = 3 / 6 = 0.5 = 50%
- Why
- A standard six-sided die has 6 equally likely outcomes. Three of them (4, 5, 6) are greater than 3, so P(A) = 3/6 = 0.5, or 50%.
- Input
- P(Ace) = 0.25 (given); Find P(not Ace)
- Result
- P(not A) = 1 − 0.25 = 0.75 = 75%
- Why
- Using the complement rule, if the probability of drawing an ace is 0.25, then the probability of NOT drawing an ace is 1 − 0.25 = 0.75, or 75%.
- Input
- P(Heart) = 0.30; P(Club) = 0.20; events are mutually exclusive
- Result
- P(Heart or Club) = 0.30 + 0.20 = 0.50 = 50%
- Why
- Hearts and clubs cannot occur simultaneously in a single draw, so the addition rule applies directly: P(A or B) = 0.30 + 0.20 = 0.50, or 50%.
- Input
- P(Heads on flip 1) = 0.5; P(Heads on flip 2) = 0.5; events are independent
- Result
- P(both Heads) = 0.5 × 0.5 = 0.25 = 25%
- Why
- Each coin flip is independent, so the joint probability is the product of the individual probabilities: P(A and B) = 0.5 × 0.5 = 0.25, or 25%.
Frequently asked questions
What is probability and how is it measured?
Probability is a measure of how likely an event is to occur. It is always expressed as a number between 0 (impossible) and 1 (certain), or equivalently as a percentage between 0% and 100%. A probability of 0.5 means the event is equally likely to happen or not happen.
What is the complement rule?
The complement rule states that the probability of an event NOT occurring equals 1 minus the probability that it does occur: P(A') = 1 − P(A). For example, if there is a 30% chance of rain, there is a 70% chance of no rain.
When can I add probabilities together?
You can add probabilities using P(A) + P(B) when the events are mutually exclusive, meaning they cannot both occur at the same time. If events can overlap, you must use the general addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
What does it mean for events to be independent?
Two events are independent when the outcome of one does not affect the outcome of the other. For example, flipping a coin twice produces independent events — the result of the first flip has no bearing on the second. Independent events use the multiplication rule: P(A ∩ B) = P(A) × P(B).
Can a probability be greater than 1?
No. A valid probability must always be between 0 and 1 inclusive. A value of 0 means an event is impossible, and a value of 1 means it is certain. Any calculation yielding a probability outside this range signals an error in the inputs.
What is the difference between theoretical and experimental probability?
Theoretical probability is calculated by reasoning about equally likely outcomes (e.g., a fair die). Experimental probability is measured by actually running trials and recording how often an event occurs. As the number of trials grows, experimental probability converges to the theoretical value — this is the Law of Large Numbers.
Sources & references
- https://www.khanacademy.org/math/statistics-probability/probability-library
- https://www.mathsisfun.com/data/probability.html
- https://openstax.org/books/statistics/pages/3-introduction
External references open in a new tab. We are independent and not affiliated with these organizations.
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- ✓ Formula and method shown above
Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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