My Sony Vaio will suddenly shut down for no apparent reason, as if the battery had just been pulled out, sometime I need to take the battery out and put it back in before that laptop will switch back on. Does the problem lie in the battery pack or in the laptop itself?

Yes, is something you can fix. All you need to do is clean up the fan vents. You will need to do some dissemblance. This video gives you an idea how to do it.

Some tips the video doesn't show.
1) Never use a magnet screwdriver. This is a great hazard for your pc.

2) Make sure you work in a table with no cloth on it as this cna have static electricity, which is VERY bad for your pc.

3) Before you start with the pc discharge yourself of any static you might have. Just touch something that's touching the ground.

Hope this helps and fixes your problem. Remember to do this every few months. I do it every 6 to keep it nice and clean. After all dust has static electricity and that's not something I want inside my computers.

Hi,

Since overheating always results in the sudden shutdown, you might solve this issue by cleaning the fans. Dust particles present in the Sony VAIO JS Series system might block the ventilation and thus the fans become unusable. By cleaning or removing those dust particles, you can resolve this overheating and thus the sudden shutdown.

Get more information to fix Shutdown Issues in Sony Laptop.

Hope this helps you.