Probability of 3 Events Calculator

How the 3-Event Probability Calculator Works

The 3-event probability calculator is a practical and essential tool for anyone working with statistics, risk analysis, or decision-making. It allows you to calculate probabilistic combinations involving three independent events, including union, intersection, and specific outcomes like "only one event", "at least two", or "no events".

Its use is straightforward: you enter the probability of each event and select the desired operation. The calculator then returns the result along with the formula used, making it ideal for both learning and applying probability rules in real-life scenarios.

Formulas Used in the Calculator

This calculator uses different formulas depending on the selected operation. For independent events A, B, and C, the main formulas are:

• Union (A ∪ B ∪ C)
  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)

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

• No events occur
  P(none) = (1 − P(A)) × (1 − P(B)) × (1 − P(C))

• Exactly one event
  P(exactly 1) = P(A)×(1−P(B))×(1−P(C)) + (1−P(A))×P(B)×(1−P(C)) + (1−P(A))×(1−P(B))×P(C)

• Exactly two events
  P(exactly 2) = P(A)×P(B)×(1−P(C)) + P(A)×(1−P(B))×P(C) + (1−P(A))×P(B)×P(C)

• At least two events
  P(≥2) = P(exactly 2) + P(A ∩ B ∩ C)

These formulas assume that all three events are independent, which is crucial for accurate results.

Practical Example of Calculation

Given:

• P(A) = 0.6

• P(B) = 0.4

• P(C) = 0.3

• Operation: Union (A ∪ B ∪ C)

Calculation:

P(A ∪ B ∪ C) = 0.6 + 0.4 + 0.3
− (0.6×0.4) − (0.6×0.3) − (0.4×0.3)

• (0.6×0.4×0.3)

Result:
P(A ∪ B ∪ C) = 0.794 or 79.4%

This represents the probability that at least one of the three events will occur.

Operation Table and Meanings

This variety of options allows you to test different probabilistic scenarios with precision.

When to Use This Calculator

• Risk analysis: estimating the chance of simultaneous failures or events

• Academic statistics: solving problems involving unions and intersections

• Decision-making: evaluating possible outcomes based on probabilities

• Uncertainty modeling: simulating real-world situations with multiple variables

It’s perfect for both students and professionals who need accurate and quick calculations.

Do the events need to be independent?

Yes. All formulas used by this calculator assume that the events are statistically independent. This means the occurrence of one event does not influence the others. If the events are dependent, different formulas are required.

Can I use this calculator for more or fewer than 3 events?

This calculator is specifically designed for three events. For more than three, formulas become more complex and need separate tools. For two events, simpler probability calculators are available.

Why use a calculator instead of manual calculation?

Although the formulas are not overly complex, manual calculations are prone to error. This calculator:

• Ensures accurate computation

• Converts results to decimals and percentages

• Lets you quickly test different probability combinations

• Displays the exact formula used for learning and transparency

It’s a reliable companion for both theory and practice.