Jim has a 5-year-old car in reasonably good condition. He wants to take out a $30,000 term (that is, accident benefit) car insurance policy until the car is 10 years old. Assume that the probability of a car having an accident in the year in which it is x years old is as follows:
x = age
5
6
7
8
9
P (accident)
0.01191
0.01292
0.01396
0.01503
0.01613
Jim is applying to a car insurance company for his car insurance policy. If the car insurance company wants to make a profit of $900 above the expected total losses, how much should it charge for the policy? Round your answer to the nearest dollar
expected payout
E[x] = 30,000(0.01191 + 0.01292 + 0.01396 + 0.01503 + 0.01613)
= $2,098.50
=> $2,099
charge needed for $900 profit
= $(2099 + 900)
= $2999
Well, Jim's car seems to be getting older and the probability of accidents is slowly increasing. But let's not get too gloomy about it.
To figure out how much the car insurance company should charge for the policy, we need to calculate the expected losses first.
Expected losses = (Probability of an accident at 5 years old * $30,000) + (Probability of an accident at 6 years old * $30,000) + (Probability of an accident at 7 years old * $30,000) + (Probability of an accident at 8 years old * $30,000) + (Probability of an accident at 9 years old * $30,000)
Expected losses = (0.01191 * $30,000) + (0.01292 * $30,000) + (0.01396 * $30,000) + (0.01503 * $30,000) + (0.01613 * $30,000)
Expected losses ≈ $1,485.90 + $1,560.00 + $1,418.80 + $1,503.90 + $1,483.90 ≈ $7,452.50
Now, the insurance company wants to make a profit of $900 on top of this. So, they should charge:
Total charge = Expected losses + $900 ≈ $7,452.50 + $900 ≈ $8,352.50
Rounding to the nearest dollar, the insurance company should charge approximately $8,353 for the policy. Keep in mind that this is just an estimate, and the actual premium may vary based on other factors.
To determine how much the car insurance company should charge for the policy, we need to calculate the expected total losses and then add the desired profit of $900.
Step 1: Calculate the expected total losses:
- Multiply the probability of having an accident in each year by the number of cars in that year.
- Multiply the result by the respective age of the car.
Expected losses at age 5 = 0.01191 * 1 * 5 = 0.05955
Expected losses at age 6 = 0.01292 * 1 * 6 = 0.07752
Expected losses at age 7 = 0.01396 * 1 * 7 = 0.09772
Expected losses at age 8 = 0.01503 * 1 * 8 = 0.12024
Expected losses at age 9 = 0.01613 * 1 * 9 = 0.14517
Total expected losses = 0.05955 + 0.07752 + 0.09772 + 0.12024 + 0.14517 = 0.5
Step 2: Add the desired profit:
Total cost = Total expected losses + Profit
Total cost = 0.5 + 900 = 900.5
Therefore, the insurance company should charge $901 to round it to the nearest dollar.
To determine the premium the car insurance company should charge, we need to calculate the expected total losses for the policy duration and add the desired profit of $900.
First, we need to calculate the expected number of accidents each year for the policy duration. We multiply the probability of having an accident in each year by the corresponding age of the car:
Expected number of accidents each year:
5 years old: 0.01191 * 5 = 0.05955
6 years old: 0.01292 * 6 = 0.07752
7 years old: 0.01396 * 7 = 0.09772
8 years old: 0.01503 * 8 = 0.12024
9 years old: 0.01613 * 9 = 0.14517
Next, we sum up the expected number of accidents for each year:
Total expected number of accidents = 0.05955 + 0.07752 + 0.09772 + 0.12024 + 0.14517 = 0.50020
Now, we can calculate the expected total losses by multiplying the expected number of accidents by the policy coverage of $30,000:
Expected total losses = 0.50020 * $30,000 = $15,006
To cover the expected total losses and make a profit of $900, the car insurance company should charge:
Premium = Expected total losses + Profit
Premium = $15,006 + $900 = $15,906
Therefore, the car insurance company should charge approximately $15,906 for the policy.