Conditional Probability Calculator

Enter the probability of event B and the probability of A and B occurring together to calculate P(A|B) — the conditional probability that event A occurs given that event B has already occurred. You can also explore P(B|A), P(A∩B), P(A∪B), and whether two events are independent. Just plug in your known values and see the full probability breakdown.

Enter a value between 0 and 1

Enter a value between 0 and 1

If left blank, assumes A and B are independent: P(A∩B) = P(A) × P(B)

Results

P(A | B) — Probability of A given B

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P(B | A) — Probability of B given A

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P(A ∩ B) — Intersection (A and B)

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P(A ∪ B) — Union (A or B)

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P(A') — Complement of A

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P(B') — Complement of B

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Are A and B Independent?

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Probability Breakdown

Frequently Asked Questions

What is conditional probability?

Conditional probability is the likelihood of an event occurring given that another event has already happened. It is written as P(A|B), meaning the probability of A given B. The formula is P(A|B) = P(A∩B) / P(B), provided P(B) > 0.

How is P(A|B) different from P(A∩B)?

P(A∩B) is the probability that both A and B occur simultaneously, while P(A|B) is the probability that A occurs knowing B has already occurred. P(A|B) adjusts the sample space to only those outcomes where B is true.

What does it mean if P(A|B) = P(A)?

If P(A|B) equals P(A), then event B has no influence on whether A occurs — the two events are statistically independent. In this case, knowing B happened tells you nothing new about the likelihood of A.

How do I calculate P(A∩B) if the events are independent?

For two independent events, the probability of both occurring is simply P(A) × P(B). Independence means one event's outcome has no effect on the other. If you leave the P(A∩B) field blank, this calculator assumes independence and computes it automatically.

What is the union of two events P(A∪B)?

P(A∪B) is the probability that at least one of the two events A or B occurs. It is calculated using the inclusion-exclusion principle: P(A∪B) = P(A) + P(B) − P(A∩B). Without subtracting the intersection, you would count it twice.

Can the probability of an event be greater than 1?

No. Probability values must always be between 0 and 1 inclusive. A value of 0 means the event is impossible, while a value of 1 means it is certain. Any result outside this range indicates an error in the input values.

What is Bayes' Theorem and how does it relate to conditional probability?

Bayes' Theorem is a formula that reverses conditional probability: P(A|B) = [P(B|A) × P(A)] / P(B). It allows you to update the probability of A based on new evidence B. This calculator derives both P(A|B) and P(B|A) directly from your inputs.

What happens if P(B) = 0?

Conditional probability P(A|B) is undefined when P(B) = 0, because you cannot condition on an event that is impossible. Mathematically, division by zero is not allowed. Make sure P(B) is greater than 0 when using this calculator.

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