It's not doing that any more -- it's no longer random or momentary. Now, if you creep up on it and surprise it, it'll keep power far enough into POST to show the BIOS splash screen before all power disappears.
It's two or three years old, and well out of warranty.
When I get home from Mother's on Monday it looks like I'll be putting the HDD into another unit to retrieve the data that's changed since Thursday's backup, but for the time being I'm on the handset only.
New laptop not due for another two or three weeks...