Question

the table shows claims and their probabilities for an insurance company. amount of claim (to the nearest $20,000) probability $0 0.70 $20,000 0.21 $40,000 0.05 $60,000 0.02 $80,000 0.01 $100,000 0.01 a. calculate the expected value. b. how much should the company charge as an average premium so that it breaks even on its claim costs? c. how much should the company charge to make a profit of $50 per policy?

Explanation:

Step1: Recall expected - value formula

The formula for the expected value $E(X)$ of a discrete random variable is $E(X)=\sum_{i}x_ip_i$, where $x_i$ are the possible values and $p_i$ are their corresponding probabilities.

Step2: Calculate the product for each row

For $x_1 = 0$ and $p_1=0.70$, the product is $0\times0.70 = 0$.
For $x_2 = 20000$ and $p_2 = 0.21$, the product is $20000\times0.21=4200$.
For $x_3 = 40000$ and $p_3 = 0.05$, the product is $40000\times0.05 = 2000$.
For $x_4 = 60000$ and $p_4 = 0.02$, the product is $60000\times0.02=1200$.
For $x_5 = 80000$ and $p_5 = 0.01$, the product is $80000\times0.01 = 800$.
For $x_6 = 100000$ and $p_6 = 0.01$, the product is $100000\times0.01=1000$.

Step3: Sum up the products

$E(X)=0 + 4200+2000 + 1200+800+1000=9200$.

Step4: Answer part b

To break - even on claim costs, the company should charge the expected value as the average premium. So the charge is $\$9200$.

Step5: Answer part c

To make a profit of $\$50$ per policy, the company should charge the expected value plus the desired profit. So the charge is $9200 + 50=\$9250$.

Answer:

a. $9200$
b. $9200$
c. $9250$