Math

Probability Calculator — Free 2026

Calculate combined, complementary, and conditional probabilities for independent or dependent events.

P(A) — Probability of Event A Enter a value between 0 and 1 Please enter a number between 0 and 1.

P(B) — Probability of Event B Enter a value between 0 and 1 Please enter a number between 0 and 1.

Event Relationship

Results

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P(A and B)

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P(A or B)

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P(not A)

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P(not B)

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P(A|B)

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P(B|A)

How It Works

Enter event probabilities
Select event relationship
Read all probability results

Understanding Probability

Probability is the mathematical framework for quantifying uncertainty. Every probability is a number between 0 (impossible) and 1 (certain). When we say the probability of rain tomorrow is 0.7, we mean that in similar conditions, it rains about 70% of the time. Understanding how to combine probabilities — when events happen together, when either can happen, or when one depends on another — is essential for decision-making in fields from medicine and engineering to finance and everyday life.

Combined Probability: AND and OR

The probability of two events both occurring (AND) depends on whether they are independent. For independent events like coin flips, P(A and B) = P(A) × P(B). The probability of flipping two heads is 0.5 × 0.5 = 0.25. For the probability of either event occurring (OR), use the addition rule: P(A or B) = P(A) + P(B) - P(A and B). The subtraction prevents double-counting the overlap. For example, the probability of drawing a heart or a king from a standard deck is 13/52 + 4/52 - 1/52 = 16/52. For related calculations with percentages, see our percentage calculator.

Conditional Probability and Bayes' Theorem

Conditional probability answers the question: "What is the probability of A, given that B has occurred?" The formula P(A|B) = P(A and B) / P(B) adjusts the probability space to only consider outcomes where B is true. This concept leads to Bayes' Theorem, one of the most powerful tools in statistics: P(A|B) = P(B|A) × P(A) / P(B). Bayes' Theorem is the foundation of spam filters, medical diagnosis, machine learning algorithms, and courtroom evidence evaluation.

Independent vs. Dependent Events

Events are independent when one does not affect the other's probability. Rolling a die and flipping a coin are independent — the die result has no bearing on the coin. Events are dependent when one outcome changes the probability of the next. Drawing cards without replacement is a classic example: after drawing an ace, the probability of drawing another ace changes from 4/52 to 3/51. Correctly identifying whether events are independent or dependent is crucial for accurate probability calculations. For more data analysis tools, explore our statistics calculator.

Frequently Asked Questions

How do I calculate P(A and B)?

For independent events: P(A and B) = P(A) × P(B). For dependent events: P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given A has occurred. Two events are independent if one occurring does not affect the probability of the other.

What is the formula for P(A or B)?

P(A or B) = P(A) + P(B) - P(A and B). This formula works for both independent and dependent events. You subtract P(A and B) to avoid double-counting the overlap where both events occur. For mutually exclusive events where P(A and B) = 0, it simplifies to P(A) + P(B).

What is conditional probability?

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. The formula is P(A|B) = P(A and B) / P(B). For independent events, P(A|B) simplifies to P(A) since B does not affect A.

What is the difference between independent and dependent events?

Independent events are those where one event does not affect the probability of the other — like flipping a coin twice. Dependent events are those where one outcome changes the probability of the next — like drawing cards without replacement. The key test: if P(A|B) = P(A), the events are independent.