Law of Total Probability
The law of total probability calculates an overall probability by splitting the population into clean groups, calculating the probability within each group, and weighting by group size.
What total probability answers
The law of total probability answers: what is the overall probability of an event when the population is split into groups? The groups must cover the whole population without overlap.
Actuarial examples are everywhere: claim probability by risk tier, lapse probability by policy duration, high-cost claim probability by age band, or flag probability by underwriting class.
Core formula
Split the population into mutually exclusive groups. For each group, multiply the probability of the event inside that group by the probability of being in that group. Then add the pieces.
Risk-class worked example
Suppose 65% of policies are standard risk and 35% are high risk. The claim rate is 4% for standard risks and 12% for high risks. The overall claim rate is 6.8%.
The calculation is not an unweighted average of 4% and 12%. The high-risk group is smaller, so its claim rate receives 35% weight.
Why Bayes needs it
Bayes theorem usually needs the total probability of the observed signal in the denominator. That denominator includes true positives and false positives.
If a flag catches high-risk policies but also flags some low-risk policies, total probability is the step that counts both paths into the flagged group.
Connection to total expectation
Total expectation is the expected-value version of the same idea. Instead of weighting conditional probabilities by group sizes, you weight conditional means by group sizes.
Exam traps
The most common error is using groups that do not cover the whole population or double-counting a group. The second is forgetting the group weights.
- Check that the groups are mutually exclusive.
- Check that the group probabilities add to 1.
- Use weighted averages, not simple averages, unless the groups are equally likely.
- For Bayes problems, include every route that can produce the observed signal.
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.
Law of Total Probability Risk-Class Drill
Original total-probability checks for risk classes, weighted claim rates, and Bayes denominators.
- Question 1/Calculation
Weighted claim rate
A portfolio is 65 percent standard risk and 35 percent high risk. Standard risks have a 4 percent claim rate, while high risks have a 12 percent claim rate. Find the overall claim rate.
Solution and grading points
The overall claim rate is 0.65 times 0.04 plus 0.35 times 0.12, which equals 0.068. The portfolio claim rate is 6.8 percent.
- Uses the class shares as weights.
- Weights each class claim rate by its class share.
- Reports 6.8 percent rather than the unweighted average of 8 percent.
- Question 2/Calculation
Overall flag rate
Three percent of policies are high risk. A flag catches 70 percent of high-risk policies and falsely flags 8 percent of other policies. What percent of all policies are flagged?
Solution and grading points
Flagged high-risk policies contribute 0.70 times 0.03, or 0.021. Flagged non-high-risk policies contribute 0.08 times 0.97, or 0.0776. The total flag rate is 9.86 percent.
- Counts both true positives and false positives.
- Uses 97 percent as the non-high-risk share.
- Adds the two routes into the flagged group.
- Question 3/Written Answer
Partition check
Before using total probability, what two checks should you make about the groups?
Solution and grading points
The groups should not overlap, and together they should cover the whole population. Their probabilities should add to 1.
- States that groups must be mutually exclusive.
- States that groups must cover the population.
- Mentions the probability-sum check.