need details solutions please

QUESTION 1 In the process of producing engine valves, the valves are subjected to a first grind. Valves whose thicknesses are within specification are ready for installation. Those valves whose thicknesses are above the specification are reground, while those whose thicknesses are below the specification are scrapped. Assume that after the first grind, 70% of the valves meet the specification, 20% are reground, and 10% are scrapped. Furthermore, assume that of those valves that are reground, 90% meet the specification, and 10% are scrapped. a) Find the probability that a valve is ground only once. (2 marks) b) Find the probability that a valve is scrapped. (2 marks) c) Given that a valve is scrapped, what is the probability that it was ground twice? (2 marks) d) Find the probability that the valve meets the specification (after either the first or second grind) (2 marks) e) Given that a valve meets the specification (after either the first or second grind), what is the probability that it was ground twice? (2 marks) Question 2 Shafts manufacturer for use in optical storage devices have diameters that are normally distributed with mean 0.652 cm and standard deviation 0.003 cm. The specification for the shaft diameter is 0.650 + 0.005 cm. ) What is the probability shaft manufactured by this process meet the specifications? (3 marks) ii) The process mean can be adjusted through calibration. If the mean is set to 0.650 cm, what is the probability the shafts will meet specification? (3 marks) iii) If the mean is set to 0.650 cm, what must the standard deviation be so that 99% of the shafts will meet specifications