Calculate compound probability for two or more events. Enter each probability as a percentage or decimal to find the chance that all or at least one occurs.

Assumes the events are independent.

Third event
Leave blank for two events. Assumes C is independent of A and B.

Related Calculators

Compound Probability Formula

The following formula is used to calculate the probability of both events A and B occurring (the intersection).

P(A cap B) & = P(A) × P(Bmid A) \ & = P(B) × P(Amid B) \ If A and B are independent: P(A cap B) & = P(A) × P(B)
  • Where P(A ∩ B) is the probability of both events A and B occurring
  • P(A) is the probability of event A
  • P(B) is the probability of event B
  • P(B|A) is the probability of event B occurring given that event A has occurred

For independent events, you can calculate the probability of both events happening by multiplying P(A) and P(B). For dependent events, use the conditional probability P(B|A) (or P(A|B)).

Compound probability for multiple independent events

Choose Many independent events and enter one probability per event. Use a percent sign for percentages or enter a decimal from 0 to 1. Separate values with commas, spaces or line breaks. Use this mode when the events are mutually independent. Knowing which of the other events occurred must leave each event’s probability unchanged.

  • All events: Multiply their probabilities, P(all) = p₁ × p₂ × … × pₙ.
  • No events: Multiply the chances that each event does not occur, P(none) = (1 − p₁) × (1 − p₂) × … × (1 − pₙ).
  • At least one event: Subtract P(none) from 1.

For independent events with chances of 50%, 40%, 25% and 20%, the chance that all occur is 0.50 × 0.40 × 0.25 × 0.20 = 1%. The chance that none occur is 0.50 × 0.60 × 0.75 × 0.80 = 18%, so the chance that at least one occurs is 82%.

Compound Probability Definition

A compound probability is the probability of two (or more) events considered together, such as “A and B” (both occur) or “A or B” (at least one occurs). For “A and B,” the probability of both events occurring is P(A ∩ B), which is found using P(A)·P(B|A). If A and B are independent, then P(B|A)=P(B) and the formula simplifies to P(A)·P(B). Probabilities can be entered as decimals or percentages as long as you use consistent units (for example, convert 25% to 0.25 when working in decimals).

How to calculate compound probability?

How to calculate compound probability?

  1. First, determine the probability of event A.

    Calculate the chance of event A occurring.

  2. Next, determine the probability of event B.

    For independent events, enter P(B). For dependent events, enter P(B given A) with P(A).

  3. Finally, calculate the compound probability.

    Calculate the compound probability using the formula above.

FAQ

What is a compound probability?

A compound probability is the probability of multiple events considered together. For example, the probability that both A and B occur is P(A ∩ B)=P(A)·P(B|A), which simplifies to P(A)·P(B) when A and B are independent.