91Ó°ÊÓ

Probability is a fundamental concept in statistics and mathematics that measures the likelihood of an event occurring. It ranges from 0 to 1, where 0 means the event will not occur and 1 means the event will certainly occur. In our exercise, we have two probabilities to consider:
1. **Survival Probability**: The probability that a 30-year-old male will survive the year, given as 0.9986.
2. **Non-Survival Probability**: The probability that the same male will not survive the year, which we calculate as 1 - 0.9986 = 0.0014. Understanding these probabilities helps us later in calculating the expected value and profit for the insurance company.

Expected value is a critical concept in both probability and statistics, representing the average outcome if an experiment is repeated many times. It's essentially the long-term average value of random variables. The formula for expected value (E) is:
\[E(X) = p_1 \times x_1 + p_2 \times x_2\]
Where:

• \(p_1\): Probability of the first event (surviving the year)
• \(x_1\): Monetary value if the first event occurs (loss of 161 dollars)
• \(p_2\): Probability of the second event (not surviving the year)
• \(x_2\): Monetary value if the second event occurs (gain of 99,839 dollars)

Using our data:
\(p_1 = 0.9986\), \(x_1 = -161\)
\(p_2 = 0.0014\), \(x_2 = 99,839\)
Substitute these values into the formula to calculate:
\[E(X) = 0.9986 \times (-161) + 0.0014 \times 99,839\]
The expected value is \(-20.989\) dollars, indicating an average loss for the 30-year-old male.

Insurance mathematics involves using mathematical models to assess and manage risks and calculate premiums, payouts, and profits. In this exercise, we use probability and expected value to evaluate policy outcomes. Here, we can look at the payout scenario:
1. **Premium Paid by Male**: When the male survives the year, he loses the insurance premium of 161 dollars. If he does not survive, the insurance pays out 100,000 dollars, less the premium paid, resulting in a 99,839 dollars payout.
2. **Company's Perspective**: For the insurance company, the expected cost per policy is calculated as:
\[0.9986 \times (-161) + 0.0014 \times 99,839 = -20.989\]
The negative expected cost indicates a profit of approximately 21 dollars per policy. This calculation is crucial for the company to set premiums that ensure profitability while offering fair payouts.

Profit calculation in insurance involves understanding both the expected premiums and payouts. For an insurance company to be profitable, the premiums collected should exceed the expected payouts. Here's a step-by-step look:
1. **Premium Collection**: The company collects 161 dollars for each policy sold.
2. **Expected Payout**: The calculated expected cost for the company is \(-20.989\) dollars, which implies that each policy sold results in an average profit of approximately 21 dollars.
Therefore, the mathematical profit for the company per policy is:
\(161 - 20.989 = 140.011\) dollars.
This profit margin ensures the sustainability of the insurance company, allowing it to cover the risks and still remain financially healthy.