Every year there is a 0.01% probability that a windmill loses a blade, which is thrown into the environment of the rotation. The damage that is caused by such an incident can cost $5,000,000. One young engineer suggests a method to install the blades in the windmill, which reduces such damage to $500,000. This new method will cost $200 per year. Should the new method be implemented? The discipline that looks at such problems is called Decision Theory. It allows us to find out the answer using matrix multiplication. Here's how it works - follow these steps:

a)

There are four possible outcomes for annual expenditure:
• Cost if the new method is not implemented and a
accident with a blade occurs: $5,000,000
• Cost if the new method is not implemented and
no accidents occur: $0
• Cost if the new method is implemented and an accident
happens: $500,200 (damage + method of maintenance)
• Cost if the new method is implemented and none
accidents happen: $200

Collect these costs in a 2-by-2 matrix where the rows correspond the possible events 'not implemented method' vs 'implemented method', and the columns correspond to the events 'accident' vs 'no accident'. This is called a utility matrix. In other words:

– the first row represents the cost if the new met-
o is not implemented, the second row represents cost
the nade if the new method is implemented;
- the first column represents the cost if an accident
occurs, the second column represents if no accidents occur.

b)

There are two possible outcomes, with different probabilities:

• an accident occurs: probability 0.0001
• no accidents occur: probability 0.9999

Collect these probabilities in a column vector. (this is called a probability vector).

c)

Multiply the matrix of costs by the probability vector. What is the size of the resulting matrix/vector?

d)

The first number in the resulting matrix/vector is through nite cost for the first year if the new method is not implemented; The second number is the average cost if the new method has been implemented. Is it best to use the new method, or continue with the old one?

e)

Another engineer suggests a cheaper method: it costs $100 per year and reduces the cost of an accident to $2,000,000. Now we have three choices: leave things as they are, implement the method to the first engineer, or implement the method to the second engineer. Try to generalize steps a)-d) to come to a decision about the three choices. What are the sizes of the matrices you need to use? What is the best decision?