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Expected value is a fundamental concept in probability that is used to predict the average outcome of a random variable over numerous trials. Think of it as the long-term mean of the random variable's possible values. It's like weighing each outcome by its chance of happening.

To calculate the expected value, you multiply each possible outcome by the probability of that outcome, then add these products together. For the life insurance policy example, the insurance company considers two scenarios:

• If the policyholder survives (probability = 0.99), the company earns the premium charged, denoted as \(C\).
• If the policyholder does not survive (probability = 0.01), the company pays out the policy amount \($75,000\), reducing its profit to \(C - 75,000\).

By computing these, the expected value \(E(X)\) helps the company determine an average outcome and is calculated as:\[E(X) = C \times 0.99 + (C - 75,000) \times 0.01 = C - 750.\]
Expected value is crucial for insurance companies to anticipate their financial performance and set appropriate policy prices.

Break-even analysis is a financial calculation used to determine the point at which the business's revenues equal its costs. Essentially, it's where there's no net gain or loss—hence the term "breaking even."

In the context of insurance, break-even analysis helps the company figure out what to charge so that they expect to make no profit or loss on average. From the calculated expected value \(E(X) = C - 750\), setting this to zero shows the charge needed to just cover losses and gains:

• Set \(C - 750 = 0\)
• Solve for \(C\): \(C = 750\)

This result means the company should charge \($750\) per policy to cover potential losses exactly without gaining or losing money. It's a safety threshold for the company, ensuring sustainability.

Determining the insurance premium is vital for insurance companies, as it balances profitability and competitiveness. The exercise illustrates how companies use probability and expected value to set premiums that align with their financial targets.

To find the needed premium for a target gain, such as a \(\(150\) profit per policy, the expected value is adjusted accordingly:

• For a net gain of \(\)150\), set the expected value to \(150\).
• Use the equation \(C - 750 = 150\).
• Solving gives: \(C = 900\).

A premium of \(\(900\) ensures the company makes an average \(\)150\) gain per policy after covering the policy payouts. By adjusting \(C\), the insurance company can control its expected returns, making careful use of statistical principles to optimize outcomes and remain competitive in the market.