How Do You Formally Test Whether Two Events Are Independent?

Two events are independent if the occurrence of one event does not change the probability of the other. Formally, test whether $P(A \cap B)=P(A)P(B)$; if the two sides are equal, the events are independent.

The Formal Tests

The following tests are equivalent when the relevant conditional probabilities are defined:

The product test is generally the safest because it remains valid even when one event has probability zero. The conditional tests follow from

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$$

If $P(A\mid B)=P(A)$, multiplying both sides by $P(B)$ gives $P(A\cap B)=P(A)P(B)$.

Worked Example

Suppose $P(A)=0.60$, $P(B)=0.50$, and $P(A\cap B)=0.30$. Then

$$P(A)P(B)=(0.60)(0.50)=0.30.$$

Since $P(A\cap B)=P(A)P(B)$, the events are independent. Alternatively,

$$P(A\mid B)=\frac{0.30}{0.50}=0.60=P(A),$$

which confirms the same conclusion.

The product compares the observed overlap with the overlap expected if neither event influences the other. Equality means knowing that one event occurred leaves the other event's probability unchanged.

For contrast, suppose $P(C)=0.40$, $P(D)=0.50$, and $P(C\cap D)=0.30$. Independence would require , but the actual intersection is , so the events are dependent. Equivalently, .

When values come from a Venn diagram, table, or tree diagram, identify both marginal probabilities and the intersection carefully. Do not round intermediate probabilities, because premature rounding can make unequal values appear equal.

A common misconception is that mutually exclusive events are independent. If both events have positive probability, mutually exclusive events have $P(A\cap B)=0$ but $P(A)P(B)>0$, so they are not independent.

IB Exam Technique

This is Mathematics: Analysis and Approaches SL 4.11, examinable on Papers 1 and 2. Show both sides of the product test numerically and state a conclusion; writing only “independent” is insufficient for show, determine, or justify.