In the process of producing engine valves, the valves are subjected to a first grind. Valves whose thicknesses are within the 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. Required:
a. Find the probability that a valve is ground only once.
b. Given that a valve is not reground, what is the probability that it is scrapped?
c. Find the probability that a valve is scrapped.
d. Given that a valve is scrapped, what is the probability that it was ground twice?
e. Find the probability that the valve meets the specification (after either the first or second grind).
f. Given that a valve meets the specification (after either the first or second grind), what is the probability that it was ground twice?
g. Given that a valve meets the specification, what is the probability that it was ground only once?