Probability sum

Description

All probabilities for mutually exclusive, exhaustive outcomes must sum to 1. This is one of the fundamental rules of probability.

$$P(A) + P(A') = 1 \implies P(A') = 1 - P(A)$$

where $A'$ is the complement of $A$ (the event that $A$ does not happen).

Example: The probability of rain on a given day is 0.35. The probability of no rain is: $$1 - 0.35 = 0.65$$

For multiple mutually exclusive outcomes: $$P(A) + P(B) + P(C) + \ldots = 1$$

This rule is often used to find a missing probability in a table or list.

Example: Outcomes of a spinner: red, blue, green with probabilities $0.3$, $x$, $0.25$. Find $x$. $$0.3 + x + 0.25 = 1 \implies x = 0.45$$

Common error: forgetting that only mutually exclusive and exhaustive events must sum to 1 — overlapping events do not follow this rule directly.

Links

Theoretical probabilities Set notation