Conditional Probability Calculator

Conditional probability measures the probability of event A given that event B has occurred, denoted mathematically as P(A|B). The foundational formula governing this relationship is derived from joint probability: P(A|B) = P(A and B) / P(B), provided that P(B) is greater than zero. In scenarios where you need to reverse or update probabilities based on new evidence—such as diagnostic testing—Bayes' Theorem is employed: P(A|B) = [P(B|A) * P(A)] / P(B). Our calculator processes your initial probabilities, marginal totals, and conditional branching values using these exact algebraic identities to instantly output your desired results.

Consider a medical screening scenario where a rare condition affects 5% of a population. Let Event A1 be 'Has Condition' (P(A1) = 0.05) and Event A2 be 'Does Not Have Condition' (P(A2) = 0.95). A diagnostic test correctly identifies the condition 90% of the time, meaning the probability of a positive test given the condition is P(B1|A1) = 0.90. However, it also yields a false positive for 10% of healthy individuals, meaning P(B1|A2) = 0.10. First, we calculate the joint probabilities: P(A1 and B1) = 0.90 * 0.05 = 0.045, and P(A2 and B1) = 0.10 * 0.95 = 0.095. Next, we sum these to find the total probability of a positive test: TotB1 = 0.045 + 0.095 = 0.140. Finally, applying the conditional formula to find the probability of actually having the condition given a positive test yields P(A1|B1) = 0.045 / 0.140 = 0.3214, or approximately 32.14%.

When working with probability inputs, always ensure that mutually exclusive events sum exactly to 1 (e.g., P(A) + P(not A) = 1). Double-check your tree diagram branches to confirm that conditional probabilities originating from the same parent node also add up to 1. A common pitfall is confusing P(A|B) with P(B|A); always verify which event is acting as the given condition before entering your parameters into the calculator.