Why do we subtract the intersection when computing the probability of the union of two events?

The probability of the intersection is calculated by rearranging the inclusion-exclusion formula to $P(E \cap F) = P(E) + P(F) - P(E \cup F)$. This is done by adding the intersection term to both sides of the original formula and subtracting the union term from both sides.

Conditions: The probabilities of the two individual events and their union are known.

The inclusion-exclusion formula for two events states that the probability of their union is the sum of their individual probabilities minus the probability of their intersection: $P(E \cup F) = P(E) + P(F) - P(E \cap F)$.

Conditions: E and F are events in the same probability space.

To solve for the intersection probability, substitute the classical probabilities (favorable outcomes divided by total outcomes) into the inclusion-exclusion formula, keep a common denominator during the algebraic steps, and isolate the unknown intersection term.

Conditions: The sample space is finite and all outcomes are equally likely.; The probabilities of the two individual events and their union are known.

The inclusion-exclusion principle should be used when considering two events that are allowed to have overlap, meaning they can occur together. It is not the relevant setup for mutually exclusive events, where the intersection is empty and the events cannot occur simultaneously.

Conditions: There are two events under consideration.; The events can occur together (their intersection is not empty).