Three percent of policies are high risk. A flag catches 70 percent of high-risk policies and falsely flags 8 percent of other policies. Among flagged policies, what percent are high risk?

High-risk flagged policies contribute 0.70 x 0.03 = 0.021. All flagged policies contribute 0.021 + 0.08 x 0.97 = 0.0986. The answer is 0.021 / 0.0986 = 0.213, or about 21.3 percent.

In a portfolio, 6 percent of policies have a claim and 1.2 percent of policies have a large claim. Among policies with a claim, what percent have a large claim?

Use the claim group as the denominator. The answer is 0.012 / 0.06 = 0.20, so 20 percent of claim policies have a large claim.

Event A has probability 30 percent, event B has probability 40 percent, and both events happen together with probability 12 percent. Are the two events independent?

Yes. The product of the two individual probabilities is 0.30 x 0.40 = 0.12, matching the joint probability. That is the independence test.

Concept

Conditional Probability

Conditional probability changes the denominator to the information you are given. It is the language behind underwriting signals, claim flags, diagnostic tests, and Bayes theorem.

What conditional probability changes

Conditional probability means the denominator becomes the event you already know happened. The question is no longer about all outcomes. It is about the outcomes inside the known condition.

That denominator shift is why conditioning is central to actuarial work. Pricing, reserving, underwriting, claim triage, and predictive models all ask how risk changes after new information appears.

Core formulas

The conditional-probability formula says to keep only the overlap with the given condition and divide by the probability of that condition. The multiplication rule is the same relationship rearranged.

Conditional probability

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

Multiplication rule

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

Claim example

Suppose 4% of policies have a claim during the year, and 25% of claim policies have a large claim. Multiplying those two numbers gives 1% of all policies with a large claim.

The 25% figure is conditional on the claim already happening. The 1% figure is unconditional across the whole portfolio. Mixing those two denominators is a common Exam P miss.

Unconditional large-claim probability

P(L\cap C)=P(L\mid C)P(C)=0.25(0.04)=0.01

Bayes and base rates

Bayes theorem reverses a conditional probability. In risk classification, that reversal is usually where the base-rate trap appears: a flag can be accurate at catching high-risk policies but still have a much lower posterior probability than intuition expects.

Example: 2% of policies are high risk. A flag catches 80% of high-risk policies and falsely flags 10% of other policies. Among flagged policies, the chance of being high risk is 0.80(0.02) divided by 0.80(0.02)+0.10(0.98), which is about 14%. The answer is not 80%.

Bayes theorem

P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}

Flag example

P(H\mid F)=\frac{0.80(0.02)}{0.80(0.02)+0.10(0.98)}\approx 0.140

Conditional distributions

For joint distributions, conditioning turns a joint probability table into a distribution over one variable after another variable is fixed. Divide each joint cell in the row or column by the marginal probability of the condition.

If you condition on X=x, every conditional probability should sum to 1 across the possible values of the other variable. That sum check catches many table errors before they become wrong answers.

Discrete conditional distribution

P(Y=y\mid X=x)=\frac{P(X=x,Y=y)}{P(X=x)}

Conditional expected value

Once you have a conditional distribution, you can take a conditional expected value. That is the bridge from probability updates to actuarial decisions: expected cost given risk class, expected severity given a claim is reported, or expected payment given a deductible applies.

Conditional expectation from a conditional distribution

E[Y\mid X=x]=\sum_y y\,P(Y=y\mid X=x)

Exam traps

Most errors come from answering the reverse conditional probability, forgetting the base rate, or using the wrong denominator.

P(A | B) and P(B | A) usually are not equal.
Independence means P(A | B)=P(A), not that A and B cannot happen together.
When a question says given, restrict the denominator to that condition.
For flags and tests, include false positives in the denominator.

Where conditional probability connects next

Bayes theorem, the law of total probability, joint distributions, conditional expectation, credibility, and classification metrics all sit on this page. If expected value is the average outcome, conditional probability is the information structure behind the average.

Practice

Original exam practice

3 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.