Of the cars that are taken to a certain garage for an M.O.T. test, \(87 \%\) pass. However, \(2 \%\) of these have faults for which they should have been failed. \(5 \%\) of the cars which fail are in fact roadworthy and should have passed.
Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random
should have passed the test, regardless of whether it actually did or not,
failed the test, given that it should have passed.
The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still \(87 \%\) overall and \(2 \%\) of the cars passed have faults as before, but now \(0.3 \%\) of the cars which should have passed are failed and \(x \%\) of the cars which are failed should have passed.