Probability & Bayes Calculator

Intuition frequently fails when evaluating probabilistic events, especially in medical screening or quality assurance where rare conditions produce counterintuitive rates of false alarms. A medical screening test with ninety-five percent accuracy can still yield far more false positives than true positives if the underlying condition prevalence across the population is extremely low. Users input prior hypothesis probabilities, true positive sensitivity rates, and false positive alarm rates to calculate updated posterior beliefs, or enter event probabilities to solve union and intersection sets.

The results display the true posterior probability given positive evidence, total marginal evidence probability, and corresponding posterior odds ratios. In multi-event modes, the tool demonstrates how compound event likelihoods behave under independence assumptions versus mutually exclusive constraints. Be careful not to confuse laboratory test sensitivity with posterior probability, as failing to account for the baseline population prior leads directly to the classic base-rate fallacy.

Bayes theorem computes posterior probability by multiplying the prior belief by the likelihood of observing the evidence, dividing by total marginal evidence across all states. For combined events, the union is calculated by summing individual probabilities and subtracting their joint intersection under probability addition rules.