Probability of 3 Events Calculator

To calculate the probabilities of three separate events (let us call them A, B, and C), the calculator applies foundational principles of set theory and probability axioms. If the events are independent, their combined metrics rely on individual decimal percentages or fractions. The core equations governing this tool include:

1. Probability of All Three Events: P(A and B and C) = P(A) × P(B) × P(C)
2. Probability of At Least One Event: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(A∩C) + P(A∩B∩C)
3. Probability of None Occurring: P(None) = 1 - P(At Least One)
4. Probability of Exactly One Event: P(Exactly One) = P(A)(1-P(B))(1-P(C)) + P(B)(1-P(A))(1-P(C)) + P(C)(1-P(A))(1-P(B))

Imagine you are managing a software development team and evaluating the likelihood of three independent system modules failing during a stress test. Let the probability of failure for Module A be 0.20 (20%), for Module B be 0.30 (30%), and for Module C be 0.50 (50%).

Step 1: Calculate the probability of all three failing.
P(All Three) = 0.20 × 0.30 × 0.50 = 0.030 or 3.0%.

Step 2: Calculate the probability of at least one module failing.
Using the inclusion-exclusion principle, we compute P(At Least One) = 0.20 + 0.30 + 0.50 - (0.20×0.30) - (0.30×0.50) - (0.20×0.50) + (0.20×0.30×0.50) = 1.00 - 0.06 - 0.15 - 0.10 + 0.03 = 0.72 or 72%.

Step 3: Calculate the probability of none failing.
P(None) = 1 - 0.72 = 0.28 or 28%.

Keep these expert tips in mind when calculating probabilities for three distinct events:

• Verify Independence: Ensure your events are truly independent before multiplying their individual probabilities directly. Dependent events require conditional probability adjustments.
• Use Decimals: Always convert percentages into decimals (e.g., 45% becomes 0.45) before inputting them into algebraic probability formulas to prevent scale errors.
• Check the Complement Rule: Remember that the probability of an event happening plus the probability of it not happening must always equal exactly 1 (or 100%).