What is joint probability?
Joint probability means the probability that A and B both occur. Marginal probabilities describe each event on its own; a conditional probability describes one event given that another occurs. Penn State’s conditional-probability lesson gives P(A and B) = P(A) × P(B given A), when the conditioning event has positive probability.
Joint probability formula and independence
For independent events, that formula becomes P(A and B) = P(A) × P(B). Penn State’s independence lesson explains the assumption: learning one event occurred does not change the other’s probability. Two events appearing on different market pages does not demonstrate independence.
Read the fictional example
The starting inputs are P(A) = 70%, P(B) = 60% and P(A and B) = 55%. The four cases are both 55%, A only 15%, B only 5% and neither 25%. They sum to 100%; adding the relevant cells returns the entered marginals. Choosing independence instead produces a 42% joint probability and cells 42%, 28%, 18% and 12%. These are constructed examples, not election forecasts.
Why joint probability has a feasible range
Both cannot be more probable than either individual event, giving the upper bound min(P(A), P(B)). The neither cell is 1 − P(A) − P(B) + P(A and B), and cannot be negative. Together with nonnegative probability, that gives the lower bound max(0, P(A) + P(B) − 1). For 70% and 60%, the joint probability can range from 30% to 60%; the marginals alone do not select one value.
How to calculate joint probability from a conditional estimate
Keep P(A) = 70% and P(B) = 60%, then choose conditional probability and enter P(B given A) = 80%. The joint result is 70% × 80% = 56%, yielding 56%, 14%, 4% and 26% across the four cases. If you enter a conditional estimate that contradicts the marginals, the calculator reports the inconsistency and removes the result. When P(A) is zero, P(B given A) is undefined; use another method.
Use it for prediction-market research
A House-and-Senate sweep is a joint event, while control of one chamber is a marginal event. The midterms guide explains why those questions and contract rules differ. The same distinction matters when reading a combo that requires several outcomes. This two-event calculator does not model larger combinations or infer dependence from prices.
Mathematical consistency is different from forecast accuracy or profitable execution. Use the Brier score calculator for a defined forecast record and the expected-value calculator to distinguish a probability estimate from trade cost.