Independent Events

Study guide: Exam P-1

Two events $A$ and $B$ are Independent if knowing that one occurred provides no information about whether the other occurred. Equivalently, they satisfy the product rule:

$P(A \cap B) = P(A) \cdot P(B)$

When $P(A) > 0$ and $P(B) > 0$ , independence means $P(A \mid B) = P(A)$ and $P(B \mid A) = P(B)$ ; an event with probability 0 is independent of every event
Independence is a symmetric relation and must be verified mathematically; it cannot be assumed from a diagram
For a collection of events $A_1, \ldots, A_n$ to be mutually independent, the product rule must hold for every subset of the collection, not just pairs:

$P(A_{i_1} \cap A_{i_2} \cap \cdots \cap A_{i_m}) = P(A_{i_1})\,P(A_{i_2}) \cdots P(A_{i_m})$

Testing Independence of Two Claim Events For two policyholders, $P(\text{A claims}) = 0.4$ , $P(\text{B claims}) = 0.3$ , and $P(\text{both claim}) = 0.12$ . Are their claim events independent?

Answer Check whether $P(A \cap B) = P(A) \cdot P(B)$ : $P(A) \cdot P(B) = 0.4 \times 0.3 = 0.12$ Since $P(A \cap B) = 0.12 = P(A) \cdot P(B)$ , the two events are independent — knowledge of one policyholder's claim does not affect the probability of the other's.