Laptop Complete Power Down....Follow

Before i begin, my battery won't hold a charge so the comp has to be plugged in or it turns off... So here's the problem.
I have a Toshiba laptop (not sure of the model but can check at some point if it really makes that much of a difference) and until recently it could play games with no problems.. i used to play WoW but stopped cause i couldn't pay internet bill.. eventually got internet back and tried to play WoW and a game called Fiesta ... now when playing either game the computer will randomly just power down with no warning but my wife can play pogo and surf myspace for 8-10 hours straight with no problems. My father thought that the problem might be lack of RAM which would be affecting the video so we went out and bought a 2 Gig stick of RAM.. but it still shuts down.. any ideas as to what might be cause'n this? would i have to upgrade video or am i in the wrong area?