Probability Calculator for 3 Independent Events — Union, Intersection, Exactly One, None

Probability Calculator for 3 Events

This probability calculator works for three independent events. Enter the probability of each event as a percentage (0–100%) or switch to decimal mode (0–1). Once you fill in P(A), P(B), and P(C), the calculator instantly outputs four results:

• P(A ∪ B ∪ C) — probability that at least one of the three events occurs (union).
• P(A ∩ B ∩ C) — probability that all three events happen simultaneously (intersection).
• P(exactly one) — probability that exactly one of the three events occurs.
• P(∅) — probability that none of the events occur.

What Are the Rules of Probability?

Here are the basic rules of probability:

• Probability takes values between 0 (impossible) and 1 (certain), inclusive.
• Complement Rule: P(A′) = 1 − P(A).
• Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Multiplication Rule (independent): P(A ∩ B) = P(A) × P(B).

Probability Rules for Three Independent Events

All Three Events (Intersection)

Multiply the individual probabilities:

P(A ∩ B ∩ C) = P(A) × P(B) × P(C)

At Least One Event (Union)

Apply the inclusion-exclusion principle:

P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
             − P(A)·P(B) − P(A)·P(C) − P(B)·P(C)
             + P(A)·P(B)·P(C)

Exactly One Event

Sum the three scenarios where only one event occurs:

P(exactly one) = P(A)·(1−P(B))·(1−P(C))
               + (1−P(A))·P(B)·(1−P(C))
               + (1−P(A))·(1−P(B))·P(C)

No Events (None)

Using the complement rule:

P(none) = 1 − P(A ∪ B ∪ C)

FAQs

How to find the probability of A and B?

For independent events, multiply their probabilities: P(A ∩ B) = P(A) × P(B). For example, if P(A) = 0.5 and P(B) = 0.3, then P(A ∩ B) = 0.15 (15%).

How to find the probability of A or B?

Use the addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For independent events this becomes P(A) + P(B) − P(A)·P(B).

What does "independent events" mean?

Events are independent if the occurrence of one does not affect the probability of the others. For example, rolling a die and flipping a coin are independent events. If events are dependent, you must use conditional probabilities instead.

Can I enter probabilities greater than 1?

No. Valid probabilities are between 0 and 1 (or 0% and 100%). The calculator will display an error if you enter values outside this range.

What is the probability of none of three events occurring?

It equals 1 minus the probability that at least one event occurs: P(none) = 1 − P(A ∪ B ∪ C). This uses the complement rule applied to the union.