Respuesta :
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) = [tex](1-0.10)^{n-1} \times0.10[/tex]
[tex]=0.90^{n-1}\times 0.10[/tex]
The probability that Actuary Toby examines more than n policies is:
P (Actuary Toby examines more than n policies) = [tex](1-0.20)^{n}[/tex]
[tex]=0.80^{n}[/tex]
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:
[tex]P(\text{Toby}>\text{Rahul})=\sum\limits^{\infty}_{n=1}{(0.90^{n-1}\times 0.10)\times 0.80^{n}}[/tex]
[tex]=\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[/tex]
Thus, the probability that Actuary Rahul examines fewer policies than Actuary Toby is 0.2857.
