When should the inclusion-exclusion principle be used instead of treating events as mutually exclusive?

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.

We subtract the intersection because adding the individual probabilities $P(E)$ and $P(F)$ counts the overlapping region twice. Subtracting $P(E \cap F)$ once corrects this double-counting so that every outcome in the union is counted exactly once.

Conditions: E and F are overlapping events in a probability space.