Probability Calculator
Solve compound probabilities, Bayes' theorem diagnostics, and binomial distributions. Convert rates to fractions and odds ratios.
Venn Diagram Overlap Visualization
Interactive Venn circles represent event space. Purple region outlines intersection set P(A ∩ B).
Step-by-step Mathematical Workings
Understanding Joint & Compound Probability
Probability represents the likelihood of an event occurring, bounded between 0 (impossible) and 1 (certain). When analyzing multiple events (Event A and Event B), their joint behavior depends on their mutual relationship:
- Independent Events: The occurrence of A does not alter the likelihood of B. Their intersection is: P(A ∩ B) = P(A) × P(B).
- Mutually Exclusive Events: The events cannot occur simultaneously. Their intersection is P(A ∩ B) = 0, and their union is simply P(A) + P(B).
- Dependent Events: The occurrence of A alters the likelihood of B. Their conditional probability is expressed via Bayes or custom intersections: P(A ∩ B) = P(A|B) × P(B).
Bayes' Theorem: Updating Beliefs
Bayes' Theorem updates the probability of a hypothesis (H) based on new evidence (E):
P(H|E) = [P(E|H) · P(H)] / P(E)
This formulation is essential in medicine (e.g. determining the probability of a disease given a positive test result) and machine learning (e.g. email spam filters).
Frequently Asked Questions About Probability
What are independent vs mutually exclusive events?
Independent events are events where the occurrence of one does not affect the probability of the other occurring (e.g. rolling a 6 on a die and flipping a heads on a coin). Mutually exclusive events are events that cannot happen at the same time (e.g. rolling a 3 and a 5 on a single die roll); their joint probability P(A and B) is exactly zero.
Why is the posterior probability of a disease often low even with a 99% accurate medical test?
This is known as the base rate fallacy. If a disease is rare (e.g., 1 in 1000 people), a test with a 5% false-positive rate will still flag about 50 healthy people out of 1000. So among all positive tests, only a small fraction (about 2%) will be true cases. Bayes' theorem takes this prior prevalence into account.
What is the Binomial Distribution?
The Binomial Distribution models the number of successes in a fixed number of independent trials (n), each with the same probability of success (p). It calculates the likelihood of getting exactly, at least, or at most k successes.
Can this apply Bayes theorem and binomial distribution?
Yes. Solve compound events, Bayes updates, unions, intersections, and binomial distributions.