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The expected value in probability and statistics is a vital concept. It serves as the average outcome one can anticipate over a series of events. Think of it as a predicted win or loss when playing a game of chance many times over. To calculate the expected value, you multiply each outcome by its probability, and then sum these products. This gives a weighted average of all possible outcomes. In the example of the construction contracts, the firm can expect a certain profit based on how the contracts are distributed. For Firm I, the expected profit is calculated by considering what profit Firm I would have in each possible scenario of contract assignments. Because each contract scenario has an equal probability of occurring, which is \( \frac{1}{9} \) for each of the nine possibilities, the expected profit is calculated by adding up the products of each potential profit outcome and its probability, resulting in: \[ E = \frac{1}{9} \times 180,000 + 4 \times \frac{1}{9} \times 90,000 + 4 \times \frac{1}{9} \times 0 = 60,000 \] This simplifies to an average profit of $60,000 for Firm I.

Random assignment is a method used to designate subjects into different groups, ensuring that each subject has an equal chance of assignment to any given group or condition. It's a critical component for maintaining fairness and objectivity in experiments and scenarios. In the case of assigning the construction contracts, random assignment means each firm has an equal chance of receiving a contract. This results in nine equally probable outcomes, as each of the three firms can potentially receive either or both of the two contracts. Random assignment ensures no firm is unfairly advantaged or disadvantaged. Having this randomness helps establish the foundations for probability calculations, such as expected value, because it means that each scenario is fair and equally likely, simplifying the calculation to just averaging out evenly across all possible outcomes.

Profit calculation is a straightforward yet essential process in business. It's the process of determining the earnings after deducting costs and expenditures. Here, we focus specifically on calculating profits based on outcomes of assigned contracts.Each contract brings in a defined profit of $90,000 for a firm. To find the total profit, you identify how many contracts a firm receives under each scenario and multiply the number by the profit from a single contract. For example, if Firm I receives both contracts (scenario (I, I)), its total profit would be \( 2 \times 90,000 = 180,000 \). If Firm I receives one contract (such as scenario (I, II)), then the profit is simply \( 90,000 \).However, if Firm I receives no contracts in a certain scenario, the profit is zero. By understanding each possible outcome's profits, businesses can make informed decisions and expectations about average profits via techniques like expected value.