(2.5 points) The probability that an airplane accident which is due to structural failure is diagnosed correctly is 0.85 and the
probability that an airplane accident which is not due to structural failure is diagnosed incorrectly as being due to structural failure is 0.35. If 30 percent of all airplane accidents are due to structural failures, find the probability that an airplane accident is due to structural failure, given that it has been diagnosed as being due to structural failure.
This is a conditional probability. The first event is the airplane accident being caused by structural failure. The probability of it being due to structural failure is 0.3 and the probability of it not being due to structural failure is 0.7. The second event involves the diagnosis of the event. If a plane fails due to structural failure, the probability that it will be diagnosed and the results will say it was due to structural failure is 0.85, and the probability that the diagnosis is unable to identify that it was because of a structural failure is 0.15. If the plane were to fail as a result of some other reason aside structural failure, the probability that the diagnosis will show that it was as a result of structural failure is 0.35 and the probability of the diagnosis showing that is is not as a result of structural failure is 0.65. To find the probability that an airplane failed due to structural failure given that it was diagnosed that it failed due to some malfunction, this is the equation;
p = (probability of plane failing and diagnosis reporting that the failure was due to structural failure)/ (probability of diagnosis reporting that failure was due to structural failure)
The two axes divide the plane into four sections called "quadrants". The quadrants are labelled with Roman numerals (not Arabic numerals), starting at the positive x-axis and going around anti-clockwise:
