Quick answer
For two events, independent events have P(A and B) = P(A) × P(B), and P(A or B) = P(A) + P(B) − P(A and B). With P(A) = 0.3 and P(B) = 0.5 independent: P(A and B) = 0.15, P(A or B) = 0.65 and P(neither) = 0.35.
This probability calculator takes the two probabilities and whether the events are independent or mutually exclusive. The calculator returns the combined probabilities.
Related tools: Implied Probability Calculator: Probability implied by betting odds. Geometric Probability Calculator: Probability the first success occurs on trial k. Dice Average Calculator: Average, minimum and maximum of rolling several dice.
How to use the Probability Calculator
- Enter p(A) (between 0 and 1).
- Enter p(B) (between 0 and 1).
- Choose events are: Independent and Mutually exclusive.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| P(A) | A number | ≥ 0, ≤ 1 |
| P(B) | A number | ≥ 0, ≤ 1 |
| Events are | Choose one: Independent; Mutually exclusive | Starts at Independent |
Formula and method
Formula
independent: P(A∩B) = P(A)P(B); exclusive: P(A∩B) = 0; P(A∪B) = P(A) + P(B) − P(A∩B)
IEEE-754 double precision; shown to at most 6 decimal places.
Exclusive events require P(A) + P(B) ≤ 1.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: P(A) = 0.3; P(B) = 0.5; Events are = Independent.
| Result | Value |
|---|---|
| P(A and B) | 0.15 |
| P(A or B) | 0.65 |
| Neither | 0.35 |
Example 2
Inputs: P(A) = 0.3; P(B) = 0.5; Events are = Mutually exclusive.
| Result | Value |
|---|---|
| P(A and B) | 0 |
| P(A or B) | 0.8 |
| Neither | 0.2 |
The first line is the headline result (p(A and B)). Below it the probability Calculator lists p(A or B) and neither, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Independent versus exclusive
Independent events do not affect each other, such as two coin flips. Mutually exclusive events cannot both happen, such as rolling a 2 and a 5 on one die, so P(A and B) = 0 and P(A or B) = P(A) + P(B).
Neither
P(neither) = 1 − P(A or B) = 1 − 0.65 = 0.35. Probabilities must lie between 0 and 1. Events that are neither independent nor exclusive need conditional probabilities, which this page does not take.
How the result changes with p
Holding every other input at the first worked example, this table changes only p. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| P | P(A and B) | P(A or B) | Neither |
|---|---|---|---|
| 0.15 | 0.075 | 0.575 | 0.425 |
| 0.3 | 0.15 | 0.65 | 0.35 |
| 0.6 | 0.3 | 0.8 | 0.2 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- P(B): entering "0.5" shows "Mutually exclusive events cannot have probabilities that add up to more than 1." beside that field and no result.
- P(A): entering "1.2" shows "P(A) must be at most 1." beside that field and no result.
Choosing between the Probability Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Implied Probability Calculator](/statistics/implied-probability/) | odds format and odds | Implied probability |
| [Geometric Probability Calculator](/statistics/geometric-probability-calculator/) | success probability and trial number | P(X = k) |
| [Dice Average Calculator](/statistics/dice-average/) | number of dice and sides per die | Average roll |
| [Permutation and Combination Calculator](/statistics/permutation-and-combination-calculator/) | total items and chosen items | Combinations (nCr) |
| [Expected Value Calculator](/statistics/expected-value/) | values and probabilities | Expected value E[X] |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- Probabilities between 0 and 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for p(B) and p(A) to confirm this.
Frequently asked questions
What is the complement of an event?
1 minus its probability.
Are coin flips independent?
Yes, each flip is unaffected by previous ones.
What if events overlap?
Use the addition rule with the overlap subtracted.
What does the Probability Calculator calculate?
Probability of A and B, and A or B, for independent or mutually exclusive events. It returns P(A and B), P(A or B) and Neither.
What formula does the Probability Calculator use?
independent: P(A∩B) = P(A)P(B); exclusive: P(A∩B) = 0; P(A∪B) = P(A) + P(B) − P(A∩B). Exclusive events require P(A) + P(B) ≤ 1.
What do I need to enter in the probability Calculator?
You enter P(A), P(B) and Events are.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Probabilities between 0 and 1.
What does the “Events are” option change?
It sets how the probability is worked out. The choices are Independent and Mutually exclusive.
What is an example of a probability result?
With p(A) 0.3, p(B) 0.5 and events are Independent, the p(A and B) is 0.15.
Related calculators
- Implied Probability Calculator — Probability implied by betting odds.
- Geometric Probability Calculator — Probability the first success occurs on trial k.
- Dice Average Calculator — Average, minimum and maximum of rolling several dice.
- Permutation and Combination Calculator — Number of permutations and combinations of r items from n.
- Expected Value Calculator — Expected value and variance of a discrete distribution.
- Bayes' Theorem Calculator — Posterior probability from a prior and test accuracy.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.