Bayes Theorem

[gwp1691]

Alain is a maintenance technician at a wind power plant, called a wind farm. There are 300 wind turbines on the wind farm. Alain’s team normally inspects about 5 turbines per day, but can inspect as many as 10. Sometimes they inspect a turbine but discover that no maintenance was actually needed. Typically, a turbine will need maintenance once every six months, but sometimes a turbine will become damaged and require emergency maintenance. Today Alain noticed a warning light on Turbine #147, an indicator that there is too much vibration in the turbine shaft. The sensors connected to the warning light are not perfectly reliable. Sometimes a turbine’s warning light will turn on for no reason at all; this happens about one day per year for each turbine, and nothing about his job is more annoying for him. Also, even if there is a problem with the turbine the warning light might fail to turn on; this happens 5% of the time when there actually is a problem with the turbine. Alain does not know when Turbine #147 was last maintained. Should Alain believe that Turbine #147 needs maintenance today?

I don't know where to even start, please give me some clues.

 

[Dr.Peterson]

Alain is a maintenance technician at a wind power plant, called a wind farm. There are 300 wind turbines on the wind farm. Alain’s team normally inspects about 5 turbines per day, but can inspect as many as 10. Sometimes they inspect a turbine but discover that no maintenance was actually needed. Typically, a turbine will need maintenance once every six months, but sometimes a turbine will become damaged and require emergency maintenance. Today Alain noticed a warning light on Turbine #147, an indicator that there is too much vibration in the turbine shaft. The sensors connected to the warning light are not perfectly reliable. Sometimes a turbine’s warning light will turn on for no reason at all; this happens about one day per year for each turbine, and nothing about his job is more annoying for him. Also, even if there is a problem with the turbine the warning light might fail to turn on; this happens 5% of the time when there actually is a problem with the turbine. Alain does not know when Turbine #147 was last maintained. Should Alain believe that Turbine #147 needs maintenance today?

I don't know where to even start, please give me some clues.

Start by making a list of (conditional) probability facts implied by the information, just as in another word problem you might list data such as lengths or speeds. Here are some such statements:

• Sometimes a turbine’s warning light will turn on for no reason at all; this happens about one day per year for each turbine.
• Also, even if there is a problem with the turbine the warning light might fail to turn on; this happens 5% of the time when there actually is a problem with the turbine.

In order to express these facts symbolically, you'll have to first define events, such as:

• N = a turbine’s warning light turns on for no reason at all.
• F = the warning light fails to turn on
• P = there actually is a problem with the turbine.

(I'm not saying these are the actual events I'd use; I'm just looking at the problem as I would initially, without having taken any steps to solve it, in order to meet you where you are. I can already see some changes I'd be making.)

Give it a try, and then start thinking about how Bayes' Theorem might apply, since that's the topic at hand. This is how problem solving starts, and the problem is clearly designed to give you practice with that.