1. ## Expected Value

An insurance company has written 59 policies of $50,000, 457 of$25,000, and 943 of $10,000 on people of age 20. If the probability that a person will die at age 20 is .001, how much can the company expect to pay during the year the policies were written? Isn't the probability that a person will die at age 20 and have a$50,000 policy $\displaystyle 0.001 \times \frac{59}{1459}$? Therefore, the expected value is $\displaystyle 0.001 \times \frac{59}{1459} \times \$50,000 + 0.001 \times \frac{457}{1459} \times \$25,000 + 0.001 \times \frac{943}{1459} \times \$10,000 = \$16.32$. My teacher disagreed, saying not to divide by 1459 (the total number of policies). She argued the expected value was $\displaystyle 59 \times \$50,000 \times 0.001 + 457 \times \$25,000 \times 0.001 + 943 \times \$10,000 \times 0.001 = \$23,805$.

A validation of either argument would be appreciated.

2. Originally Posted by NOX Andrew
An insurance company has written 59 policies of $50,000, 457 of$25,000, and 943 of $10,000 on people of age 20. If the probability that a person will die at age 20 is .001, how much can the company expect to pay during the year the policies were written? Isn't the probability that a person will die at age 20 and have a$50,000 policy $\displaystyle 0.001 \times \frac{59}{1459}$? Therefore, the expected value is $\displaystyle 0.001 \times \frac{59}{1459} \times \$50,000 + 0.001 \times \frac{457}{1459} \times \$25,000 + 0.001 \times \frac{943}{1459} \times \$10,000 = \$16.32$. My teacher disagreed, saying not to divide by 1459 (the total number of policies). She argued the expected value was $\displaystyle 59 \times \$50,000 \times 0.001 + 457 \times \$25,000 \times 0.001 + 943 \times \$10,000 \times 0.001 = \$23,805$.

A validation of either argument would be appreciated.
Your teacher is correct. There is a probability of 0.01 that 59 policies have to pay out $50, 000, a probability of 0.01 that 457 policies have to pay out$25, 000 and a probability 0.01 that 943 policies have to pay out \$10, 000.