What's the Formula for \(P(A \cup B)\)?

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It calculates the probability that at least one of the events occurs: event $A$, event $B$, or both.

Why Is the Intersection Subtracted?

The symbol union, $A \cup B$, means all outcomes belonging to $A$ or $B$, including outcomes in both events. If you simply calculate $P(A)+P(B)$, outcomes in the intersection, $A \cap B$, are counted twice.

Subtracting $P(A \cap B)$ once corrects this double counting. This is called the addition rule of probability and belongs to IB Mathematics: Applications and Interpretation subtopic SL 4.6, combined events and Venn diagrams.

For example, suppose

Then

Therefore, the probability that $A$ or $B$, or both, occurs is $0.60$.

To see the same mechanism as frequencies, imagine 100 students. Forty study French, 35 study Spanish, and 15 study both languages. Adding 40 and 35 gives 75, but the 15 bilingual students have been included twice. Subtracting that overlap gives $40+35-15=60$, so 60 students study at least one language. Dividing by 100 gives the same probability, $0.60$.

A useful rearrangement is $P(A \cap B)=P(A)+P(B)-P(A \cup B)$. This lets you find the overlap when the union is known. Always check that your final probability lies between $0$ and $1$ and is at least as large as each individual probability. For instance, $P(A \cup B)$ cannot be smaller than $P(A)$ because every outcome in $A$ is already in the union.

A common misconception is that mutually exclusive and independent mean the same thing. They do not: mutually exclusive events cannot occur together, while independent events can occur together but do not affect each other's probabilities.

IB Exam Technique

In an IB response, write the general formula before substituting values. If a Venn diagram or table is given, identify the overlap carefully; only omit the intersection term when the events are explicitly mutually exclusive.