Question

the table shows claims and their probabilities for an insurance company. use the table to answer parts a through c.

amount of claim (to the nearest $50,000) | probability
--- | ---
$0 | 0.65
$50,000 | 0.21
$100,000 | 0.07
$150,000 | 0.05
$200,000 | 0.01
$250,000 | 0.01

a. calculate the expected value.
$ 29500

b. how much should the company charge as an average premium so that it breaks even on its claim costs?
$\square

Explanation:

Part a: Calculate the expected value

Step 1: Recall the formula for expected value

The expected value \( E(X) \) of a discrete random variable is calculated as \( E(X)=\sum_{i} x_i P(x_i) \), where \( x_i \) is the value of the random variable and \( P(x_i) \) is the corresponding probability.

Step 2: Multiply each claim amount by its probability

• For \( x = 0 \), \( 0\times0.65 = 0 \)
• For \( x = 50000 \), \( 50000\times0.21 = 10500 \)
• For \( x = 100000 \), \( 100000\times0.07 = 7000 \)
• For \( x = 150000 \), \( 150000\times0.05 = 7500 \)
• For \( x = 200000 \), \( 200000\times0.01 = 2000 \)
• For \( x = 250000 \), \( 250000\times0.01 = 2500 \)

Step 3: Sum the products

\( E(X)=0 + 10500+7000 + 7500+2000 + 2500=29500 \)

Step 1: Understand break - even for insurance

To break even on claim costs, the average premium charged should be equal to the expected value of the claim amount. This is because the expected value represents the average claim cost per policy.

Step 2: Set the premium equal to the expected value

Since the expected claim cost (from part a) is \( \$29500 \), the company should charge a premium equal to this expected value to break even.

Answer:

\( \$29500 \)

Part b: Determine the break - even premium