Probability Calculator

Calculate probabilities of events with this free online calculator. Find union, intersection, and other related probabilities of two independent events.

Probability of Two Events

Enter the probabilities of two independent events to calculate various probability relationships between them.

Probability of A: P(A)

Probability of B: P(B)

Probability Description	Value
Probability of A NOT occurring: P(A')	?
Probability of B NOT occurring: P(B')	?
Probability of A and B both occurring: P(A∩B)	?
Probability that A or B or both occur: P(A∪B)	?
Probability that A or B occurs but NOT both: P(AΔB)	?
Probability of neither A nor B occurring: P((A∪B)')	?
Probability of A occurring but NOT B: P(A∩B')	?
Probability of B occurring but NOT A: P(B∩A')	?

Key Probability Formulas

\(P(A') = 1 - P(A)\) - Complement of A

\(P(A \cap B) = P(A) \times P(B)\) - Intersection (for independent events)

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) - Union

\(P(A \Delta B) = P(A) + P(B) - 2 \times P(A \cap B)\) - Symmetric Difference

Example

If P(A) = 0.6 and P(B) = 0.3:

P(A') = 1 - 0.6 = 0.4
P(A∩B) = 0.6 × 0.3 = 0.18
P(A∪B) = 0.6 + 0.3 - 0.18 = 0.72
P(AΔB) = 0.6 + 0.3 - 2 × 0.18 = 0.54

About Probability

Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, which is expressed as a number between 0 and 1.

Basic Probability Concepts

Sample Space (S): The set of all possible outcomes of an experiment.
Event: A subset of the sample space.
Probability of an Event: The number of favorable outcomes divided by the total number of possible outcomes.
Complement of an Event: All outcomes in the sample space that are not in the event.

Probability Rules

Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
Multiplication Rule (Independent Events): P(A and B) = P(A) × P(B)
Complement Rule: P(not A) = 1 - P(A)

Types of Events

Independent Events: The occurrence of one event does not affect the probability of the other event.
Dependent Events: The occurrence of one event affects the probability of the other event.
Mutually Exclusive Events: Two events cannot occur simultaneously.

Applications of Probability

Statistics: Used in hypothesis testing, confidence intervals, and data analysis.
Finance: Used in risk assessment, portfolio management, and option pricing.
Insurance: Used to calculate premiums based on the likelihood of claims.
Medicine: Used in clinical trials, diagnostic testing, and epidemiology.
Computer Science: Used in machine learning, artificial intelligence, and cryptography.