The liquid crystal display in the new Extime brand of digital watches is such that the probability it continues to function for at least \(x\) hours before failure is constant (for any given choice of the number \(x\) ), regardless of how long the display has already been functioning. The expected value of the number of hours the display functions before failure is known to be 30,000 .

(a) Define the appropriate probability space for the experiment of observing the number of hours a display of the type described above functions before failure.

(b) What is the probability that the display functions for at least 20,000 hours?

(c) If the display has already functioned for 10,000 hours, what is the probability that it will continue to function for at least another 20,000 hours?

(d) The display has a rather unique guarantee in the sense that any purchaser of a Extime watch, whether the watch is new or used, has a warranty on the display of 2 years from the date of purchase, during which time if the display fails, it will be replaced free of charge. Assuming that the number of hours the watch operates in a given period of time is essentially the same for all buyers of the Extime watch, is it more likely that a buyer of a used watch will be obtaining a free display replacement than a buyer of a new watch, given an equal period of watch ownership? Explain.