Determine the expected cost of operating a large facility for two years, given the following information: The operating costs are $450,000 annually for high demand (probability 0.7... Determine the expected cost of operating a large facility for two years, given the following information: The operating costs are $450,000 annually for high demand (probability 0.7) and $300,000 annually for low demand (probability 0.3).
Understand the Problem
The question asks us to calculate the expected operating cost of the large facility over two years, considering the probabilities of high and low demand. We need to calculate a weighted average of the operating costs under each demand scenario.
Answer
\$1,650,000
Answer for screen readers
$1,650,000
Steps to Solve
- Calculate the operating cost for the first year under high demand.
Under high demand, the operating cost for the first year is $900,000.
- Calculate the operating cost for the first year under low demand.
Under low demand, the operating cost for the first year is $700,000.
- Calculate the expected operating cost for the first year.
The probability of high demand is 60% (0.60) and low demand is 40% (0.40). The expected operating cost is calculated as follows:
$$ \text{Expected Cost Year 1} = (0.60 \times $900,000) + (0.40 \times $700,000) $$
$$ \text{Expected Cost Year 1} = $540,000 + $280,000 = $820,000 $$
- Calculate the operating cost for the second year under high demand, given high demand in the first year.
Given high demand in the first year, the operating cost in the second year will be $950,000.
- Calculate the operating cost for the second year under low demand, given high demand in the first year.
Given high demand in the first year, the operating cost in the second year, if demand is low, will be $750,000.
- Calculate the expected operating cost for the second year, given high demand in the first year.
The probability of high demand in the second year, given high demand in the first year, is 80% (0.80), and the probability of low demand is 20% (0.20). The expected operating cost is calculated as follows:
$$ \text{Expected Cost Year 2 | High Demand Year 1} = (0.80 \times $950,000) + (0.20 \times $750,000) $$
$$ \text{Expected Cost Year 2 | High Demand Year 1} = $760,000 + $150,000 = $910,000 $$
- Calculate the operating cost for the second year under high demand, given low demand in the first year.
Given low demand in the first year, the operating cost in the second year, if demand is high, will be $850,000.
- Calculate the operating cost for the second year under low demand, given low demand in the first year.
Given low demand in the first year, the operating cost in the second year will be $650,000.
- Calculate the expected operating cost for the second year, given low demand in the first year.
The probability of high demand in the second year, given low demand in the first year, is 30% (0.30), and the probability of low demand is 70% (0.70). The expected operating cost is calculated as follows:
$$ \text{Expected Cost Year 2 | Low Demand Year 1} = (0.30 \times $850,000) + (0.70 \times $650,000) $$
$$ \text{Expected Cost Year 2 | Low Demand Year 1} = $255,000 + $455,000 = $710,000 $$
- Calculate the overall expected operating cost for the second year.
To calculate the expected operating cost for the second year, we need to weight the expected cost given high demand in the first year and the expected cost given low demand in the first year, by their respective initial probabilities:
$$ \text{Expected Cost Year 2} = (0.60 \times $910,000) + (0.40 \times $710,000) $$
$$ \text{Expected Cost Year 2} = $546,000 + $284,000 = $830,000 $$
- Calculate the total expected operating cost over two years.
$$ \text{Total Expected Cost} = \text{Expected Cost Year 1} + \text{Expected Cost Year 2} $$
$$ \text{Total Expected Cost} = $820,000 + $830,000 = $1,650,000 $$
$1,650,000
More Information
The expected operating cost for the facility over the two-year period is calculated by considering the probabilities of high and low demand in each year and their impact on the operating costs. The initial demand affects the subsequent year's expected costs, leading to a weighted average calculation.
Tips
A common mistake would be to not consider the conditional probabilities. For example, one might simply average all high demand and low demand costs and therefore ignore that the probability for the second year is dependent on what happened in the first year.
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