The board of directors of a company knows that the probability that carbon emissions from the company’s factory exceed the permissible level is 35%. They hire a consultant who uses a carbon footprint calculator to test the emissions level. The test, which has an accuracy rate of 85%, indicates that the factory's carbon emissions are within the permissible level.

Given the test result, the probability that carbon emissions from the factory are actually within the permissible level is ......?

Question

The board of directors of a company knows that the probability that carbon emissions from the company’s factory exceed the permissible level is 35%. They hire a consultant who uses a carbon footprint calculator to test the emissions level. The test, which has an accuracy rate of 85%, indicates that the factory's carbon emissions are within the permissible level.

Given the test result, the probability that carbon emissions from the factory are actually within the permissible level is ......?

anonymous · Answer

i think we can do this if we do not get lost
my method is to forget about working with the decimals and picking some actual number to start with, say they have 10,000 trials, of which 35% or 3500 exceed the level

anonymous · Answer

you test with 85% accuracy so of the 3500 that are bad, 85% of the, or $.85\times 3500=2979$ test correctly as bad, whereas 15% of the remaining 6500 also test bad, or $.15*6500=975$ test bad

anonymous · Answer

the total number that test bad are $2979+975=3957$ and the total that are actually bad are $2979$ so your probability is $$\frac{2979}{3957}$$

anonymous · Answer

you can do it without this crutch of picking a number $$\frac{.85\times .35}{.85\times .35+.15\times .65}$$ will give you the same answer