The Alliance Corp expects to sell a number of copper cable units at varying prices under three economic scenarios. Each scenario's probability is as follows. What is the expected value of the total sales projections? - Scenario A: Probability - 0.30, Units - 200, Price - $15 - Scenario B: Probability - 0.50, Units - 320, Price - $30 - Scenario C: Probability - 0.20, Units - 410, Price - $40

To find the expected value of the total sales projections, we need to calculate the expected sales for each scenario and then sum them together.

Here's how we can calculate it:

For Scenario A: Probability (P_A) = 0.30 Units (U_A) = 200 Price (Pr_A) = $15 Expected Sales (ES_A) = P_A * U_A * Pr_A

For Scenario B: Probability (P_B) = 0.50 Units (U_B) = 320 Price (Pr_B) = $30 Expected Sales (ES_B) = P_B * U_B * Pr_B

For Scenario C: Probability (P_C) = 0.20 Units (U_C) = 410 Price (Pr_C) = $40 Expected Sales (ES_C) = P_C * U_C * Pr_C

Now, let's plug in the numbers to find the expected sales for each scenario:

ES_A = 0.30 * 200 * $15 = $900 ES_B = 0.50 * 320 * $30 = $4800 ES_C = 0.20 * 410 * $40 = $3280

Finally, we sum up the expected sales from all scenarios to get the total expected value of sales projections:

Total Expected Value (TEV) = ES_A + ES_B + ES_C TEV = $900 + $4800 + $3280 = $8980

So, the expected value of the total sales projections is $8980.

Extra: The concept of expected value is very important in probability and statistics. It is a weighted average of all possible values of a random variable, with the weights being their respective probabilities. In business, expected value calculations can help in decision-making under uncertainty by analyzing different outcomes based on various probabilities.

In this particular problem, we're evaluating expected sales under three different scenarios with associated probabilities, which is a common approach in financial and business forecasting. By considering different scenarios, a company can plan for various potential futures and make informed decisions on inventory, staffing, and budgeting based on these projections.

Understanding expected value is also foundational to more complex topics in economics, finance, and insurance, such as risk management and actuarial science, where calculating the expected outcomes can help in assessing risks and premiums. It is also a key concept in game theory, decision theory, and various other fields that involve uncertainty and decision-making under risk.