What Does It Mean for Two Events to Be Independent?

Two events are independent if the occurrence of one event does not change the probability of the other. For events $A$ and $B$, the main test is $P(A \cap B)=P(A)P(B)$.

This belongs to SL 4.6: Venn diagrams, combined events and conditional probability, which is studied at both SL and HL.

The Reasoning

Suppose $A$ is the event that a fair coin lands heads and $B$ is the event that a fair six-sided die shows an even number.

$$P(A)=\frac{1}{2}, \qquad P(B)=\frac{3}{6}=\frac{1}{2}.$$

The probability that both events occur is

$$P(A \cap B)=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}.$$

Because $P(A \cap B)=P(A)P(B)$, the events are independent. Knowing that the coin landed heads gives no information about whether the die result was even.

For events with non-zero probabilities, independence can also be expressed using conditional probability:

$$P(A\mid B)=P(A)$$

or equivalently

$$P(B\mid A)=P(B).$$

In words, knowing that $B$ occurred does not alter the probability of $A$.

Consider a second example. A card is drawn from a standard deck, replaced, and the deck is shuffled before another card is drawn. Let $A$ be “the first card is an ace” and $B$ be “the second card is an ace”. Replacement keeps $P(B)=\frac{4}{52}$ whether or not occurred, so the events are independent. Without replacement, while , so they are dependent. This shows that independence comes from unchanged probabilities, not simply from events having different names.

A common misconception is that independent means “cannot happen together.” That describes mutually exclusive events. Two mutually exclusive events with positive probabilities are not independent because observing one tells you that the other did not occur.

Exam Technique

In an IB Math AI response, show the relevant probabilities and explicitly verify whether $P(A \cap B)=P(A)P(B)$. Do not state independence merely because events appear unrelated; justify it numerically. This can be tested in either Paper 1 or Paper 2, both of which are GDC-active.