Conditional probability and independence

2 free worked examples

Solved in full, free to everyone, and never rotated. Read these before deciding whether the rest of the drill is worth paying for.

1. Exam PConditional probability and independenceCore

   Among all claims, 30% are commercial and 18% of all claims are both commercial and above 10,000. Given that a claim is commercial, what is the probability it exceeds 10,000?

   1. A$0.1800$
   2. B$0.3000$
   3. C$0.4200$
   4. D$0.5400$
   5. E$0.6000$

   Solution

   1. Let $A$ be 'above 10,000' and $B$ be 'commercial'. The 18% is a JOINT probability out of all claims.
   2. $P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0.18}{0.30}$.
   3. $=0.60$.
   4. The conditional probability is larger than the joint one because the denominator has shrunk to the commercial claims only.

   Trap. Reading the 18% as already conditional on being commercial.

2. Exam PConditional probability and independenceExam level

   Of a group of policyholders, 40% own a home, 25% own a boat, and 15% own both. Given a policyholder owns a home, what is the probability they also own a boat?

   1. A$0.1500$
   2. B$0.2500$
   3. C$0.3750$
   4. D$0.5000$
   5. E$0.6000$

   Solution

   1. The conditioning event is 'owns a home', so $P(\text{home})=0.40$ is the denominator.
   2. $P(\text{boat}\mid \text{home})=\frac{P(\text{both})}{P(\text{home})}=\frac{0.15}{0.40}$.
   3. $=0.375$.
   4. Since $P(\text{boat})=0.25\ne 0.375$, home and boat ownership are NOT independent in this book - owning a home raises the chance of owning a boat.