Calculate the probability of single events, union of events, intersection, conditional probability P(A|B), and complements.
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A probability calculator computes the mathematical likelihood of an outcome occurring within a defined sample space. Ranging from 0 (impossible) to 1 (certainty), probability models everything from coin tosses and insurance risk underwriting to quantum mechanics and predictive artificial intelligence.
P(A) = \frac{n(A)}{n(S)}, \quad P(A \cup B) = P(A) + P(B) - P(A \cap B), \quad P(A|B) = \frac{P(A \cap B)}{P(B)}Where n(A) is the count of favorable outcomes, n(S) is total sample space outcomes, P(A ∪ B) is the probability of either event occurring, and P(A|B) is the conditional probability of A given B has occurred.
Scenario: Calculate the probability of rolling a 2, 4, or 6 on a single fair 6-sided die.
Identify favorable outcomes: {2, 4, 6} → n(A) = 3.
Identify total possible outcomes: {1, 2, 3, 4, 5, 6} → n(S) = 6.
Apply the classical probability formula: P(A) = 3 / 6 = 0.50 (50.00%).
Determine odds: 1 to 1 (even odds).
Mutually exclusive events cannot happen at the same time (e.g., flipping heads and tails on a single coin flip). Independent events do not influence each other’s likelihood (e.g., flipping heads on coin #1 and rolling a 6 on die #2).
To convert probability P into odds in favor, divide P by (1 - P). For example, a 25% probability (0.25) gives odds in favor of 0.25 / 0.75 = 1 to 3 (1:3).
Conditional probability measures the probability of event A occurring given that event B is already known to have occurred. It shrinks the effective sample space to outcome B.
The law of total probability states that if events B1, B2, ..., Bn partition a sample space, the total probability of an event A is the sum of P(A|Bi) × P(Bi) across all partitions.
Calculate conditional posterior probability P(A|B), positive predictive value (PPV), false positives, and medical test accuracy.
Calculate combinations nCr and permutations nPr with or without repetition, factorial expansions, and subset ordering.
Compute exact P(X=k) and cumulative binomial probabilities P(X≤k), P(X≥k), distribution mean, variance, and standard deviation.