Conditional Probability

Study guide: Exam P-1

Conditional Probability $P(A \mid B)$ is the probability that event $A$ occurs given that event $B$ is known to have occurred, restricting the sample space to $B$ .

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$

If $A$ and $B$ are independent, then $P(A \mid B) = P(A)$
Conditional probability is foundational to Bayes Theorem, The Law of Total Probability, and various insurance and actuarial calculations

Claim Severity Given Deductible Threshold A loss $X$ is uniformly distributed on $(0, 1000)$ . Given that the loss exceeds 400, what is the probability it also exceeds 700?

Answer Let $A = \{X > 700\}$ and $B = \{X > 400\}$ . Since $A \subseteq B$ , we have $A \cap B = A$ . $P(A \mid B) = \frac{P(X > 700)}{P(X > 400)} = \frac{300/1000}{600/1000} = \frac{300}{600} = 0.5$