Calculate single and multiple event probabilities easily.
Calculate probability of single events or combine multiple events.
Probability P(A)
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As percentage: — · Odds: —
The Probability Calculator handles two scenarios: single-event probability (favorable outcomes divided by total outcomes) and multiple-event probability (combining two events with AND or OR logic). The calculator handles independent events, dependent events (with conditional probability), and mutually exclusive events — covering the most common cases you will encounter in introductory probability and statistics.
P(A) = Favorable outcomes ÷ Total outcomes
P(A ∩ B) = P(A) × P(B)
P(A ∩ B) = P(A) × P(B|A)
Where P(B|A) is the conditional probability of B given that A has occurred.
P(A ∪ B) = P(A) + P(B)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The subtraction prevents double-counting the overlap.
Single event: What is the probability of drawing a heart from a standard deck? Favorable = 13 (hearts), Total = 52. P = 13/52 = 0.25 (25%).
Independent AND: Probability of flipping two heads in a row? P(H) = 0.5 each. P(H and H) = 0.5 × 0.5 = 0.25.
Dependent AND: Two cards drawn without replacement. P(first is heart) = 13/52. P(second is heart | first was heart) = 12/51. P(both hearts) = (13/52) × (12/51) ≈ 0.0588.
Mutually exclusive OR: P(rolling 1 or 2 on a die) = 1/6 + 1/6 = 1/3.
Non-exclusive OR: P(heart or king) = 13/52 + 4/52 − 1/52 = 16/52 ≈ 0.308. (The king of hearts is in both, so we subtract one to avoid double-counting.)
Probability is the mathematical foundation of statistics, risk assessment, and decision-making under uncertainty. Every weather forecast ("30% chance of rain"), medical test ("95% accurate"), insurance premium, and investment decision relies on probability calculations. Understanding probability helps you spot misleading statistics, evaluate risk intelligently, and make better decisions when outcomes are uncertain.
Two concepts deserve special attention. First, independence: two events are independent if the occurrence of one does not affect the probability of the other. Coin flips are independent; drawing cards without replacement is not. Second, conditional probability: P(B|A) is the probability of B given that A has occurred. This is the basis of Bayes\' theorem, which powers modern spam filters, medical diagnosis, and machine learning. Mastering these two concepts gives you the tools to reason about uncertainty in any domain.
Probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). For equally likely outcomes: P(event) = (favorable outcomes) ÷ (total outcomes).
If A and B are independent: P(A and B) = P(A) × P(B). If dependent: P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given that A occurred.
If A and B are mutually exclusive: P(A or B) = P(A) + P(B). If not: P(A or B) = P(A) + P(B) − P(A and B). The subtraction avoids double-counting.
P(B|A) is the probability of B given that A has occurred. Formula: P(B|A) = P(A and B) ÷ P(A). Used in Bayes' theorem and medical testing.
P(not A) = 1 − P(A). Useful when it is easier to calculate the probability of an event not happening. Example: P(not rolling a 6) = 1 − 1/6 = 5/6.
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