Independent events and unions of events

• Determine or justify whether two events $A$ and $B$ are independent using probabilities or a two-way table.
• Calculate the probability of an intersection $P(A \cap B)$ when events are given as independent, including repeated independent trials (such as “at least one” problems).
• Calculate the probability of a union $P(A \cup B)$ using the general addition rule.
• Distinguish between independent events and mutually exclusive (disjoint) events.

Core ideas

Independent events

Two events $A$ and $B$ are independent if knowing that one event has occurred does not change the probability that the other event occurs. Assuming $P(A) > 0$ and $P(B) > 0$, you can verify independence using any of these three equivalent checks:

1. $P(A \mid B) = P(A)$
2. $P(B \mid A) = P(B)$
3. $P(A \cap B) = P(A) \cdot P(B)$

If the two sides are equal, the events are independent; if they are not equal, the events are dependent.

Multiplication rule for independent events

When (and only when) you know or have proven that $A$ and $B$ are independent, the general multiplication rule simplifies to:

Already on Pro? Sign in

• Read the free lesson: Statistical studies, investigative questions and variables
• Practise this topic10 free questions a day