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In the exercise, the concept of a probability distribution is crucial. Probability distribution describes how the probabilities are distributed over different possible outcomes. It gives a mathematical description of a random phenomenon in terms of its sample space and probabilities.

For the given problem, the random variable \(X\) representing the damage during a certain type of accident has specific possible values: \(0, \)1000, \(5000, and \)10000. These values are each associated with a respective probability of occurring: 0.8 for no damage, 0.1 for \(1000, 0.08 for \)5000, and 0.02 for $10000. This setup allows the expected damage (average outcome based on probabilities) to be calculated.

Through the probability distribution, you can better understand and anticipate the likelihood of different levels of damage. This is an important foundation for calculating risk in fields like insurance.

Insurance deductible plays a significant role in determining how much an insurance company will pay when a damage claim is filed. In this context, a deductible is an amount the insurer does not cover, which must come out of the policyholder's pocket.

For example, in this exercise, we have a $500 deductible. When calculating payouts, anything up to that $500 doesn't require the insurer to contribute. This means if damage is equal to or less than $500, the company pays nothing.

For situations where the damage exceeds the deductible, the payment from the insurer would be the damage amount minus $500. This helps reduce the insurer's risk and encourages careful behavior from the insured party, as they have to cover any damage up to the deductible themselves.

Expected profit is the amount a company anticipates earning from a transaction, accounting for all probabilities and expenses. Calculating expected profit requires a balance between the premium charged for a service and the potential costs incurred.

In the exercise, to find the expected profit, we need to determine how the premium (the payment made by the insured to the insurer) compares against the expected payouts.

The company plans for an expected profit of $100. This means they aim not just to cover the expected payouts but also to have a surplus of $100. To reach this profit goal from a mix of possible damage outcomes, the calculation involved subtracting the expected payout, $600, from the desired total earnings, which includes the target profit.

Thus, the required premium would be $700, ensuring the company meets both expected claims and its profit target. This mathematical foresight is crucial for financial stability and planning in businesses like insurance.