12. Consider a manufacturing process composed of three stages - (R1, R2 and R3). Stage one

consists of 2 machines arranged in a doubly redundant sub system. Stage two has 2

machines that work in series, meaning that this stage operates successfully only if all these

machines operate. Stage three is made up of 4 machines and only operates successfully if a

least 1 such machine operates. The three manufacturing stages are connected in series to

produce the final product, meaning that they must all work properly to produce a final

product. Between two scheduled maintenances, the two machines at stage 1 (R1) are not

independent of each other. Whilst the probability of machine 1 failing is 0.20 and the

probability of machine 2 failing is 0.10, the probability of machine 1 failing given machine!

has failed is 0.68. Between two scheduled maintenances, all the other machines may fail

independently of the others, with the following probabilities:- Stage (R2): 0.25 0.45,

Stage (R3): 0.44 0.21 0.15 0.54. Use this information to answer the following

questions to 3 decimal places (and expressing probability as a number between 0 and 1).

The reliability of sub system stage 1 is

2. The probability of sub system stage 2 failing is

3. The reliability of sub system stage 3 is

4. At stage 3 the probability of exactly 3 machines breaking down between scheduled

maintenances is

5. The probability of the manufacturing process as a whole working between scheduled