Probability Calculator

Calculate probabilities for independent events, series of events, and normal distributions.

▼ Enter the values and click Calculate

Probability of Two Events

Find the union, intersection, and other related probabilities of two independent events.

Probability of A: P(A)

Probability of B: P(B)

P(A') —

P(B') —

P(A ∩ B) —

P(A ∪ B) —

P(A Δ B) —

P((A ∪ B)') —

Probability Solver for Two Events

Provide any two known probability values below to calculate the remaining probabilities of two independent events.

Probability of A: P(A)

Probability of B: P(B)

Probability of A NOT occurring: P(A')

Probability of B NOT occurring: P(B')

Probability of A and B: P(A ∩ B)

Probability that A or B or both occur: P(A ∪ B)

Probability that A or B occurs but NOT both: P(A Δ B)

Probability of neither A nor B occurring: P((A ∪ B)')

What Is Probability?

Probability is a measure of how likely an event is to occur. It is expressed as a value between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.

Basic Probability:
P(A) = Number of favorable outcomes ÷ Total number of possible outcomes

For two events, probability can also be used to calculate the probability that both events occur, either event occurs, or neither event occurs.

Basic Probability Rules

Complement:
P(A') = 1 − P(A)

Intersection:
P(A ∩ B) = Probability that both A and B occur

Union:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Neither A nor B:
P((A ∪ B)') = 1 − P(A ∪ B)

Exclusive OR:
P(A Δ B) = P(A) + P(B) − 2P(A ∩ B)

How to Use the Probability Calculator

Step 1: Enter the known probability values.

Step 2: Make sure every probability is between 0 and 1.

Step 3: Click Calculate.

Step 4: Review the calculated probability values.

Frequently Asked Questions

What is probability?

Probability measures how likely an event is to occur. Probability values range from 0 to 1.

What does a probability of 0 mean?

A probability of 0 means that the event is impossible.

What does a probability of 1 mean?

A probability of 1 means that the event is certain to occur.

Can probability be greater than 1?

No. A probability must always be between 0 and 1, inclusive.

What is the complement of an event?

The complement is the probability that the event does not occur. It is calculated using P(A') = 1 − P(A).

What is the probability of A or B occurring?

For two events, the union is calculated using P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

What is the probability of neither A nor B occurring?

It is the complement of the probability that A or B occurs: P((A ∪ B)') = 1 − P(A ∪ B).

Disclaimer

This calculator is provided for general mathematical and educational purposes. While we aim to provide accurate results, SmartMyTools does not guarantee that the results are suitable for professional, academic, financial, engineering, or other specialized applications.