Both HID headlight bulbs out on '01 CL-S

So my daughter called me last week to tell me that the driver's bulb was out on her '01 CL-S. As I was getting ready to order bulbs, she texted me tonight to tell me that her other headlight is now out.

Given that these bulbs are over 14 years old, is it possible in this situation that the bulbs are simply shot and not the ballasts that are often at fault?

Also, can I use a Sylvania HID replacement like this one?

Sylvania HID Bulb D2R: Advance Auto Parts