Reverse conditional probability

A question is this type if and only if it requires finding P(earlier stage|later stage), working backwards through the tree using Bayes' theorem or conditional probability.

3 questions

OCR S1 2009 January Q8
8 A game uses an unbiased die with faces numbered 1 to 6 . The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
  1. Find the probability that the final score is 4 .
  2. Given that the die is thrown only once, find the probability that the final score is 4 .
  3. Given that the die is thrown twice, find the probability that the final score is 4 .
Edexcel S1 2018 June Q3
  1. A manufacturer of electric generators buys engines for its generators from three companies, \(R , S\) and \(T\).
Company \(R\) supplies 40\% of the engines. Company \(S\) supplies \(25 \%\) of the engines. The rest of the engines are supplied by company \(T\). It is known that \(2 \%\) of the engines supplied by company \(R\) are faulty, \(1 \%\) of the engines supplied by company \(S\) are faulty and \(2 \%\) of the engines supplied by company \(T\) are faulty. An engine is chosen at random.
  1. Draw a tree diagram to show all the possible outcomes and the associated probabilities.
  2. Calculate the probability that the engine is from company \(R\) and is not faulty.
  3. Calculate the probability that the engine is faulty. Given that the engine is faulty,
  4. find the probability that the engine did not come from company \(S\).
Edexcel S1 Q6
  1. 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
    1. should have passed the test, regardless of whether it actually did or not,
    2. 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.
  2. Find the value of \(x\).