Question

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 policies are mutually independent. Calculate the probability that Actuary Rahul examines fewer policies than Actuary Toby.

Answer 1

Answer:

The probability that Actuary Rahul examines fewer policies than Actuary Toby is 0.2857.

Step-by-step explanation:

It is provided that the 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.

The probability that a low-risk policy has a claim is, P (L) = 0.10.

The probability that a high-risk policy has a claim is, P (H) = 0.20.

For positive integer n, the probability that Actuary Rahul examines exactly n policies is:

P (Actuary Rahul examines exactly n policies) = $(1-0.10)^{n-1} \times0.10$

                                                                             $=0.90^{n-1}\times 0.10$

The probability that Actuary Toby examines more than n policies is:

P (Actuary Toby examines more than n policies) = $(1-0.20)^{n}$

                                                                                 $=0.80^{n}$

It is provided that  the claim statuses of policies are mutually independent.

Compute the probability that Actuary Toby examines more policies than Actuary Rahul as follows:

$P(\text{Toby}>\text{Rahul})=\sum\limits^{\infty}_{n=1}{(0.90^{n-1}\times 0.10)\times 0.80^{n}}$

                           $=\frac{0.10}{0.90}\sum\limits^{\infty}_{n=1}{0.90^{n}\times 0.80^{n}}\\\\=0.1111\times \sum\limits^{\infty}_{n=1}{0.72^{n}}\\\\=0.1111\times \frac{0.72}{1-0.72}\\\\=0.285686\\\\\approx 0.2857$

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