Independent events

Definition

Independent events are two or more events where the occurrence of one event does not affect the probability of the other events occurring. Mathematically, events A and B are independent if $P(A \cap B) = P(A) \cdot P(B)$.

5 Must Know Facts For Your Next Test

1. For two independent events A and B, $P(A|B) = P(A)$ and $P(B|A) = P(B)$.
2. The independence of two events implies that knowing the outcome of one provides no information about the other.
3. If events A and B are independent, then their complements (A' and B') are also independent.
4. In a sequence of trials, like flipping a fair coin multiple times, each flip is an independent event.
5. Independence is different from mutual exclusivity; mutually exclusive events cannot happen simultaneously, whereas independent events can.

Review Questions

• What condition must be true for two events to be considered independent?
• How does independence differ from mutual exclusivity in terms of probability?
• If you know that $P(A \cap B) = P(A) \cdot P(B)$, what can you conclude about events A and B?

Related terms

Mutually Exclusive Events: Events that cannot occur at the same time. If one event happens, the other cannot.

Conditional Probability: The probability of an event occurring given that another event has already occurred, denoted as $P(A|B)$.

Complementary Events: Two mutually exclusive events whose probabilities sum to one. If A is an event, its complement is denoted as A'.