Life insurance A life insurance company sells a term...

Life insurance A life insurance company sells a term insurance policy to a 21 -year-old male that pays $\$ 100,000$ if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $\$ 250$ each year as payment for the insurance. The amount $Y$ that the company earns on this policy is $\$ 250$ per year, less the $\$ 100,000$ that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$ profit earned by the sompany: $$ \begin{array}{lccccc} \hline \text { Age at death: } & 21 & 22 & 23 & 24 & 25 & 26 \text { or more } \\ \text { Profit: } & -\$ 99,750 & -\$ 99,500 & -\$ 99,250 & -\$ 99,000 & -598,750 & \$ 1250 \\ \text { Probability: } & 0.00183 & 0.00186 & 0.00189 & 0.00191 & 0.00193 & \\ \hline \end{array} $$ (a) Explain why the company suffers a loss of $\$ 98,750$ on such a policy if a client dies at age 25 . (b) Find the missing probability. Show your work. (c) Calculate the mean $\mu_{Y}$. Interpret this value in context.

Life insurance A life insurance company sells a term insurance policy to a 21 -year-old male that pays $\$ 100,000$ if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $\$ 250$ each year as payment for the insurance. The amount $Y$ that the company earns on this policy is $\$ 250$ per year, less the $\$ 100,000$ that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$ profit earned by the sompany:
$$
\begin{array}{lccccc}
\hline \text { Age at death: } & 21 & 22 & 23 & 24 & 25 & 26 \text { or more } \\
\text { Profit: } & -\$ 99,750 & -\$ 99,500 & -\$ 99,250 & -\$ 99,000 & -598,750 & \$ 1250 \\
\text { Probability: } & 0.00183 & 0.00186 & 0.00189 & 0.00191 & 0.00193 & \\
\hline
\end{array}
$$
(a) Explain why the company suffers a loss of $\$ 98,750$ on such a policy if a client dies at age 25 .
(b) Find the missing probability. Show your work.
(c) Calculate the mean $\mu_{Y}$. Interpret this value in context.

Key Concepts

Discrete Random Variable

This concept refers to a variable that can take a countable set of outcomes, each associated with a specific probability. In insurance, the profit or loss from a policy is modeled as a discrete random variable whose outcomes depend on when an event, such as death, occurs.

Expected Value

The expected value is the weighted average of all possible outcomes of a random variable, with weights given by their respective probabilities. In the context of insurance, calculating the expected profit or loss helps the company determine the fairness and profitability of the policy over time.

Total Probability Rule

The total probability rule states that the sum of the probabilities of all mutually exclusive outcomes must equal one. It is a fundamental principle used to find missing probabilities by subtracting the sum of the known probabilities from one.

Insurance Profit and Loss Calculation

This concept involves computing the net gain or loss for the insurer by subtracting the payout cost from the premiums received. It reflects how the timing of a claim, such as the insured's death, affects the overall financial outcome of the policy.

Mortality Tables

Mortality tables provide estimated probabilities of death at various ages and are crucial in the insurance industry. They are used to assess risk and determine appropriate premiums, ensuring that the probability of a claim is accurately reflected in the pricing of the policy.

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