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Probability Calculator

Calculate intersections, unions, and complements of independent events with visual Venn diagrams.

Two Events
Probability Solver
Series of Events

Enter probabilities between 0 and 1.

Please provide any 2 values below to calculate the remaining probabilities of two independent events.

Calculate the probability of repeated independent events occurring over multiple trials.

Event A

Event B

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Understanding Independent Probability

Probability is the mathematical measure of how likely an event is to occur. When analyzing multiple events, the core rule of Independent Probability states that the outcome of one event does not affect the outcome of another. Flipping a coin and rolling a die are independent events—the coin landing on heads does not change the odds of the die landing on a six.

Core Probability Formulas (Independent Events)

Using standard set theory notation, we define two independent events as $A$ and $B$. Their probabilities are written as $P(A)$ and $P(B)$. The foundational formulas used in this calculator are:

  • Complement (NOT occurring): $P(A') = 1 - P(A)$
  • Intersection (A AND B): $P(A \cap B) = P(A) \times P(B)$
  • Union (A OR B or both): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
  • Exclusive OR (A OR B but NOT both): $P(A \Delta B) = P(A \cup B) - P(A \cap B)$
  • Neither A nor B: $P((A \cup B)') = 1 - P(A \cup B)$

Venn Diagrams

Venn diagrams provide a visual representation of how these probabilities overlap. In our calculator, the left circle represents Event A and the right circle represents Event B. The colored segments directly correspond to the mathematical outputs:

  • The Overlap (Intersection): Represents $P(A \cap B)$, the exact scenario where both events occur simultaneously.
  • The Entire Figure (Union): Represents $P(A \cup B)$, capturing all scenarios where at least one of the events occurs.
  • Outside the Circles: Represents the probability that neither event occurs.

Series of Independent Events (Repeated Trials)

When you repeat an independent event multiple times, the probability scales exponentially. If the probability of Event A occurring once is $P(A)$, then the exact probability of it occurring $n$ times in a row is calculated as $(P(A))^n$.

Conversely, calculating the probability of an event occurring at least once in a series is mathematically intensive to calculate directly. Instead, statisticians calculate the probability of it never occurring $(1 - P(A))^n$, and subtract that from $1$. The formula is: $1 - (1 - P(A))^n$.