What is the addition rule for probability?

The addition rule finds the probability that event A or event B (or both) occurs: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If the events are mutually exclusive (they can't both happen), the overlap term drops out: P(A ∪ B) = P(A) + P(B).

What is conditional probability?

Conditional probability, P(A|B), is the probability that event A occurs given that event B has already occurred. It's calculated as P(A ∩ B) / P(B), and requires P(B) to be greater than zero.

What is Bayes' Theorem used for?

Bayes' Theorem lets you 'reverse' a conditional probability. If you know P(B|A) — how likely B is given A — you can find P(A|B) — how likely A is given B — as long as you also know P(A) and P(B|¬A).

Why do I need P(B|¬A) for Bayes' Theorem?

P(B|¬A) is needed to compute the overall probability of B occurring across both cases (when A happens and when it doesn't), which is the denominator of Bayes' formula. Without it, there's no way to normalize the result into a valid probability.

Can probabilities be entered as percentages instead of decimals?

Enter probabilities as decimals between 0 and 1 (0.25 instead of 25%). The result is displayed as a percentage for readability, but inputs must be in decimal form.

What does it mean if two events are mutually exclusive?

Mutually exclusive events cannot occur at the same time — for example, flipping a coin can't land on both heads and tails at once. When events are mutually exclusive, P(A ∩ B) = 0, simplifying the addition rule.

Why did I get an error saying P(B) is zero?

Conditional probability and Bayes' Theorem both divide by P(B). If P(B) is zero — meaning B is impossible under your inputs — the conditional probability is mathematically undefined, so the calculator shows an error instead of a misleading answer.

Where is Bayes' Theorem used in real life?

Bayes' Theorem is the foundation of medical test interpretation (given a positive test, what's the real chance of having the condition?), spam filtering, and machine learning classifiers.

Probability Calculator — Addition Rule, Conditional Probability, and Bayes' Theorem

Our Probability Calculator covers three of the most commonly needed probability formulas in one tool. Use the Addition Rule to find P(A ∪ B) for two events (with a mutually-exclusive shortcut), Conditional Probability to find P(A|B) from a joint and marginal probability, or Bayes' Theorem to reverse a conditional probability when you know P(B|A) but need P(A|B). Every mode shows the formula, your substituted values, and the final result as a percentage.

Quick Answer

This calculator covers three probability formulas: the Addition Rule (P(A ∪ B) = P(A) + P(B) − P(A ∩ B)), Conditional Probability (P(A|B) = P(A ∩ B) / P(B)), and Bayes' Theorem, which reverses a conditional probability. Pick a mode, enter your known probabilities as decimals, and get an instant percentage result.

P(A)

P(B)

Events are mutually exclusive

P(A ∩ B)

Enter the known probabilities, then click Calculate.

How to Use the Probability Calculator — Addition Rule, Conditional & Bayes' Theorem

1
Choose a mode: Addition Rule, Conditional Probability, or Bayes' Theorem.
2
Enter the known probabilities as decimals between 0 and 1 (e.g., 0.25 for 25%).
3
For the Addition Rule, check 'mutually exclusive' if the two events can never happen together.
4
Click 'Calculate' to see the result as a percentage along with the step-by-step formula substitution.

Why Use Probability Calculator — Addition Rule, Conditional & Bayes' Theorem?

Probability problems trip people up not because the formulas are hard to state, but because it's easy to plug values into the wrong formula or forget a term — especially with Bayes' Theorem, where P(A|B) and P(B|A) are easy to confuse despite being very different quantities. Our calculator keeps the three formulas cleanly separated into dedicated modes and shows the substituted formula alongside the result, so you can verify your own probability homework or get a quick, trustworthy answer for a real-world estimation problem.

Frequently Asked Questions

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