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In probability theory, a probability distribution describes how the values of a random variable are distributed. It provides a detailed map of how often we can expect each possible outcome to occur over many trials or observations.
In the context of the given exercise, the random variable is the damage amount, denoted as \(X\), in an accident. The possible values \(X\) can assume are \(0, 1000, 5000,\) and \(10000\) dollars, with corresponding probabilities of \(0.8, 0.1, 0.08,\) and \(0.02\). This means:

• There is an 80% chance of no damage.
• A 10% chance of \(\\(1000\) in damages.
• An 8% chance of \(\\)5000\) in damages.
• A 2% chance of \(\$10000\) in damages.

Understanding probability distribution allows insurers to anticipate potential losses and set premiums accordingly by using principles like expected value (or expectation), which is a measure of the central tendency of the distribution.

Insurance deductibles represent the amount a policyholder must pay out of pocket before the insurance company begins to cover the costs. Deductibles help mitigate risk for the insurer by discouraging small claims and sharing the liability with the policyholder.
In our exercise, a \(\\(500\) deductible is implemented. This affects how the insurance company calculates its payouts. For instance:

• If the damage is \(\\)1000\), the insurance covers \(\\(1000 - \\)500 = \\(500\).
• For \(\\)5000\) in damages, the company pays \(\\(4500\).
• And for \(\\)10000\) in damages, the payout is \(\$9500\).

The deductible only influences payouts higher than its own value. Thus, when determining profit strategies, insurance companies must consider these adjusted payouts instead of the initial damage amounts.

Expected profit is a financial metric used to calculate what an insurance company expects to gain from a policy, considering potential payouts and premiums collected. To compute it, it's essential to understand both the expected payout and the desired profit margin.
The formula for the expected profit in insurance terms is:\[\text{Expected Profit} = \text{Premium Collected} - E[Payout]\]In this textbook problem, the insurance company wishes to achieve an expected profit of \(\\(100\). The expected payout, calculated based on adjusted payouts from damage amounts, is \(\\)600\).
To solve for the necessary premium, set up the equation:\[\text{Premium} - 600 = 100\]This equation suggests that the insurance company must charge a premium of \(\$700\) to reach its profit goal. This ensures that even after potential payouts, the company maintains a steady profit margin.