
Conditional Probability
The probability that an airplane accident that is due to structural failure is correctly diagnosed is 0.85, and the probability that an airplane accident that is not due to structural failure is diagnosed as being structural failure is 0.35. If 0.3 percent of all airplane accidents are structural failure, what is the probability that an airplane accident is due to structural failure given that it has been diagnosed as die to structural failure.
Let X ~ airplane accents and Y ~ structure failure
$\displaystyle P(X \cap Y) = 0.85 \ P(X \cap Y^c) = 0.35 \ P(Y) = 0.3 \ P(Y^c) = 0.7$
$\displaystyle P(XY) = \frac{P(X \cap Y)P(Y)}{P(X \cap Y)P(Y) + P(X \cap Y^c)P(Y^c)}$
Is that the correct formula to use to solve this problem?

Let $\displaystyle A_1$ be the event that the structural failure is the reason of accident
let $\displaystyle A_2$ be the event that there is other reason of accident than structure failure.
let $\displaystyle X$ be the event that airplane accident is diagnosed due to structural failure...
then $\displaystyle P(A_1)=0.3\;\;\;;\;\;P(A_2)=0.7\;\;\;;\;P(XA_1)=0 .85\;\;\;;\;P(XA_2)=0.35$
with the above, Use Baye's Theorem to find $\displaystyle P(A_1X)$

Also, just in case a picture helps...
http://www.ballooncalculus.org/draw/prob/condd.png
... where G = a good plane, B = a bad (structurally failing) plane, P = diagnosed positively for structural failure, N = negatively.
Numbers as follows: 0.3% means it might be wise to multiply 85 and 15 both by 3, making 300, from which we get the other 99.7% as simply 99,700; this in turn shared 65 : 35.
Then you can see that the fraction of positives that are true positives is in fact (only) 255 / (255 + 29,910).
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