Question

4. Automobile policies are separated into two groups: low-risk and high-risk. Actuary Rahul examines low-risk policies, continuing until a policy with a claim is found and then stopping. Actuary Toby follows the same procedure with high-risk policies. Each low-risk policy has a 10% probability of having a claim. Each high-risk policy has a 20% probability of having a claim. The claim statuses of polices are mutually independent. Calculate the probability that Actuary Rahul examines fewer policies than Actuary Toby.

Answer 1

Answer:

Step-by-step explanation:

for n ∈ N

Since Actuary Rahul examines low-risk policies, continuing until a policy with a claim is found and then stopping.

∴ the probability that Actuary Rahul examines exactly n policies

$(0.9)^{n-1}.(0.1)---(1)$

the probability that Actuary Toby examines more than  exactly n policies

$(0.8)^n---(2)$

Given that policies are actually independent

∴ the probability that the event  (1) and (2) happens simultaneously is

$(0.9)^{n-1}*(0.1)*(0.8)^n$

∴  the probability that Actuary Rahul examines fewer policies than Actuary Toby

$\sum ^\infty _{n=1} (0.9)^{n-1}*(0.1)*(0.8)^n\\\\=(\frac{0.1}{0.9} \sum ^\infty _{n=1}(0.72)^n\\\\=\frac{1}{9} (\frac{0.72}{0.28} )\\\\=\frac{2}{7} \\\\=0.2857$

the probability that Actuary Rahul examines fewer policies than Actuary Toby is 0.2857