Determine the expected cost of operating a large facility for two years given the following information: The cost to build the large facility is $2 million. If demand is high, oper... Determine the expected cost of operating a large facility for two years given the following information: The cost to build the large facility is $2 million. If demand is high, operating costs are $450,000 annually. If demand is low, operating costs are $300,000. The likelihood of high demand is 0.7 and the likelihood of low demand is 0.3.
Understand the Problem
The question asks us to calculate the expected operating cost of the large facility over two years. This involves weighting the operating costs under high and low demand scenarios by their respective probabilities and then multiplying the result by 2 (for the two years).
Answer
$148,000
Answer for screen readers
$148,000
Steps to Solve
- Calculate the expected operating cost for the first year
The expected operating cost is the sum of the operating costs in each scenario, weighted by the probability of that scenario occurring:
$$ \text{Expected Cost} = (\text{Cost}\text{High Demand} \cdot \text{Probability}\text{High Demand}) + (\text{Cost}\text{Low Demand} \cdot \text{Probability}\text{Low Demand}) $$
Substitute the given values:
$$ \text{Expected Cost} = ($90,000 \cdot 0.6) + ($50,000 \cdot 0.4) $$
- Simplify the equation
Multiply the costs by their probabilities:
$$ \text{Expected Cost} = $54,000 + $20,000 $$
- Calculate the total expected cost for one year
Add the values from the previous step:
$$ \text{Expected Cost} = $74,000 $$
- Calculate the expected operating cost over two years
Multiply the expected operating cost for one year by 2:
$$ \text{Total Expected Cost} = $74,000 \cdot 2 $$
- Calculate the final answer
$$ \text{Total Expected Cost} = $148,000 $$
$148,000
More Information
The expected value is a fundamental concept in probability and statistics. It represents the average outcome you would expect if you repeated an experiment or situation many times. In this case, it's the average operating cost we expect over many instances of the same facility operating under the given demand probabilities.
Tips
A common mistake is forgetting to multiply the one-year expected cost by the number of years (in this case, 2). Another mistake would be calculating the weighted average incorrectly, such as not multiplying the operating costs by their corresponding probabilities.
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