probability 3 events calculator

How this probability 3 events calculator works

This tool helps you calculate probabilities involving three events: A, B, and C. Depending on your data, you can use one of three approaches: independent events, dependent events with intersections, or conditional chain rule.

When to use each mode

1) Independent mode

Use this when occurrence of one event does not change the probability of the others. Example: three unrelated system checks where each check succeeds independently.

• All three: P(A ∩ B ∩ C) = P(A)P(B)P(C)
• At least one: 1 - (1-P(A))(1-P(B))(1-P(C))
• Exactly one and exactly two are computed directly from combinations.

2) Dependent mode (inclusion-exclusion)

Use this when events overlap and are not independent. You provide single-event probabilities plus pairwise and triple intersections.

• P(A ∪ B ∪ C) = P(A)+P(B)+P(C)-P(A∩B)-P(A∩C)-P(B∩C)+P(A∩B∩C)
• None = 1 - P(A ∪ B ∪ C)
• Exactly one and exactly two come from overlap decomposition.

3) Conditional mode (chain rule)

Use this when you know a sequence: first A, then B given A, then C given A and B. This is common in reliability pipelines, multi-step qualification funnels, and Bayesian setups.

• P(A ∩ B ∩ C) = P(A) × P(B|A) × P(C|A∩B)

Quick example

Suppose events are independent with P(A)=0.6, P(B)=0.5, and P(C)=0.2. Then the chance all three happen is 0.6×0.5×0.2 = 0.06 (6%). The chance at least one happens is 1-(0.4×0.5×0.8)=0.84 (84%).

Common mistakes to avoid

• Mixing decimal and percent formats incorrectly.
• Assuming independence when events clearly influence each other.
• Entering inconsistent overlap values (e.g., P(A∩B) larger than P(A)).
• Forgetting that final probabilities must be between 0 and 1.

Why this matters

Three-event probability appears in finance, medical testing, quality control, operations research, machine learning evaluation, and risk management. Having a calculator that supports independent, dependent, and conditional structures helps you model real-world systems more accurately.