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

$\displaystyle P(X \cap Y) = 0.85 \ P(X \cap Y^c) = 0.35 \ P(Y) = 0.3 \ P(Y^c) = 0.7$

$\displaystyle P(X|Y) = \frac{P(X \cap Y)P(Y)}{P(X \cap Y)P(Y) + P(X \cap Y^c)P(Y^c)}$

Is that the correct formula to use to solve this problem?