Conditional Probability Calculator

What Is Conditional Probability?

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is written as $P(A|B)$ and read as "the probability of A given B."

Conditional Probability Formula

The fundamental formula for conditional probability is:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Where $P(A \cap B)$ is the joint probability of both events occurring and $P(B)$ is the probability of the conditioning event.

How to Use This Calculator

Enter $P(A)$, $P(B|A)$, and $P(B|\bar{A})$ (the probability of B given A does not occur). The calculator builds the full probability tree and computes all joint and conditional probabilities, including $P(A|B)$, $P(A|\bar{B})$, and their complements.

Real-World Example

Suppose 5% of a population has a disease. A test has 91% sensitivity (true positive rate) and 95% specificity (true negative rate). If a person tests positive, what is the probability they actually have the disease? Using conditional probability, $P(D|T+) \approx 48.9\%$, showing that a positive test alone does not guarantee infection.

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