Question
An insurance company charges a customer $1600 per year for a particular customer's auto insurance. The company has predicted that there is a 10% chance the person will make a claim on the policy of $5000 (which means the insurance company would lose $3400) and 90% chance that they won't make a claim.
What can the insurance company on average expect to make on selling this policy?
Question 10 options:
$2900
$940
$1100
$1020
What can the insurance company on average expect to make on selling this policy?
Question 10 options:
$2900
$940
$1100
$1020
Answers
GPT-4o mini
To calculate the expected income from the insurance policy, we can use the formula for expected value (EV):
\[
EV = (P(\text{no claim}) \times \text{profit if no claim}) + (P(\text{claim}) \times \text{profit if claim})
\]
Where:
- $P(\text{no claim})$ is the probability of no claim,
- $P(\text{claim})$ is the probability of a claim,
- \text{profit if no claim} is the profit made if no claim is made,
- \text{profit if claim} is the profit made if a claim is made.
From the problem:
- The insurance company charges the customer $1600 per year.
- There is a 10% chance (0.10) that the person will make a claim, which costs the company $5000. Therefore, the company's loss if a claim is made is $5000 - $1600 = -$3400 (the customer makes the claim and the insurance pays $5000, resulting in a loss of $3400).
- There is a 90% chance (0.90) that the person will not make a claim, so the profit in this case is simply $1600 (no payout).
Calculating the expected value:
1. **Profit if no claim is made (90% probability)**:
\[
P(\text{no claim}) = 0.90
\]
\[
\text{Profit if no claim} = 1600
\]
2. **Profit if claim is made (10% probability)**:
\[
P(\text{claim}) = 0.10
\]
\[
\text{Profit if claim} = -3400
\]
Now plug values into the EV formula:
\[
EV = (0.90 \times 1600) + (0.10 \times -3400)
\]
Calculating each term:
- For no claim:
\[
0.90 \times 1600 = 1440
\]
- For claim:
\[
0.10 \times -3400 = -340
\]
Now, sum these values:
\[
EV = 1440 - 340 = 1100
\]
Therefore, on average, the insurance company can expect to make **$1100** on selling this policy.
So the answer is:
**$1100**
\[
EV = (P(\text{no claim}) \times \text{profit if no claim}) + (P(\text{claim}) \times \text{profit if claim})
\]
Where:
- $P(\text{no claim})$ is the probability of no claim,
- $P(\text{claim})$ is the probability of a claim,
- \text{profit if no claim} is the profit made if no claim is made,
- \text{profit if claim} is the profit made if a claim is made.
From the problem:
- The insurance company charges the customer $1600 per year.
- There is a 10% chance (0.10) that the person will make a claim, which costs the company $5000. Therefore, the company's loss if a claim is made is $5000 - $1600 = -$3400 (the customer makes the claim and the insurance pays $5000, resulting in a loss of $3400).
- There is a 90% chance (0.90) that the person will not make a claim, so the profit in this case is simply $1600 (no payout).
Calculating the expected value:
1. **Profit if no claim is made (90% probability)**:
\[
P(\text{no claim}) = 0.90
\]
\[
\text{Profit if no claim} = 1600
\]
2. **Profit if claim is made (10% probability)**:
\[
P(\text{claim}) = 0.10
\]
\[
\text{Profit if claim} = -3400
\]
Now plug values into the EV formula:
\[
EV = (0.90 \times 1600) + (0.10 \times -3400)
\]
Calculating each term:
- For no claim:
\[
0.90 \times 1600 = 1440
\]
- For claim:
\[
0.10 \times -3400 = -340
\]
Now, sum these values:
\[
EV = 1440 - 340 = 1100
\]
Therefore, on average, the insurance company can expect to make **$1100** on selling this policy.
So the answer is:
**$1100**