In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final color and intensity of light are to be met. Let X and Y denote the thickness of two different layers of ink It is known that X is normally distributed with a mean of 0.1 mm and a standard deviation of 0.00031 mm, and Y is also normally distributed with a mean of 0.23 mm and a standard deviation of 0.00017 mm. Assume that these variables are independent.

a. If a particular lamp is made up of these two inks only, what is the probability that the total ink thickness is less than 0.2337 mm?
b. A lamp with a total ink thickness exceeding 0.2405 mm lacks the uniformity of color that the customer demands. Find the probability that a randomly selected lamp fails to meet customer specifications.