Question

Automobile policies are separated into two groups: low-risk and high-risk. Actuary Rahul examines low-risk policies, connuing unl 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:

The probability that Actuary Rahul examines fewer policies that Actuary Toby = 0.2857

Step-by-step explanation:

It is said that Actuary Rahul examines a low risk policy

Probability of a low risk policy having a claim = 10% = 0.1

Actuary Toby examines high risk policy

Probability of a high risk policy having a claim = 20% = 0.2

Let the number of policies examined by actuary Rahul before he finds a claim and stop be n

Probability that actuary Rahul examines exactly n policies =  $0.9^{n-1} (0.1)$

Probability that Toby examines more than n policies = $0.8^n$

Since the claim statuses of policies are mutually independent, the probability that both events happen simultaneously = $0.9^{n-1} (0.1) (0.8)^n$

probability that both events happen simultaneously = $\frac{0.1}{0.9} (0.72^{n})$

The probability that Actuary Rahul examines fewer policies that Actuary Toby = $\sum\limits^ \infty_1 {\frac{0.1}{0.9} 0.72^{n} }$ = $\frac{1}{9}\sum\limits^ \infty_1 { 0.72^{n} } = \frac{1}{9} (\frac{0.72}{1-0.72} ) = \frac{1}{9} (\frac{0.72}{0.28} )$

The probability that Actuary Rahul examines fewer policies that Actuary Toby = 0.2857