Bayes' theorem and the law of total probability

2 free worked examples

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1. Exam PBayes' theorem and the law of total probabilityCore

   A motor book is 50% low risk, 30% standard and 20% high risk, with annual claim probabilities 2%, 5% and 11%. What proportion of policyholders claim in a year?

   1. A$0.0180$
   2. B$0.0350$
   3. C$0.0470$
   4. D$0.0600$
   5. E$0.0900$

   Solution

   1. This is the law of total probability across a three-part partition.
   2. $P(\text{claim})=0.5(0.02)+0.3(0.05)+0.2(0.11)$.
   3. $=0.010+0.015+0.022$.
   4. $=0.047$. This figure is the denominator of every Bayes calculation on this book, so it is worth computing once and reusing.

   Trap. Averaging 2%, 5% and 11% without weights, which gives 6% and assumes equal class sizes.

2. Exam PBayes' theorem and the law of total probabilityCore

   Three suppliers provide 25%, 35% and 40% of parts, with failure rates 0.02, 0.04 and 0.05. What is the overall failure rate?

   1. A$0.0300$
   2. B$0.0367$
   3. C$0.0390$
   4. D$0.0410$
   5. E$0.0500$

   Solution

   1. Weight each supplier's failure rate by its share of parts.
   2. $0.25(0.02)+0.35(0.04)+0.40(0.05)$.
   3. $=0.005+0.014+0.020$.
   4. $=0.039$, between the smallest and largest rates as any weighted average must be.