An LCD screen cannot produce enough light by itself. It needs a Backlight to help illuminate the screen.
Power for the Backlight is produced by an Inverter. An Inverter also provides power for the LCD screen.
It reduces voltage for the Backlight, and increases Voltage for the LCD screen, from the laptop's power.
Most of the time a dark LCD screen can be attributed directly to a bad Inverter. (Around 90 percent)
A Backlight is a CCFL.
Cold Cathode Fluorescent Light.
VERY similar to the fluorescent lights you see used in residential, and commercial applications.
(Homes, and businesses)
Only on a MUCH smaller scale in physical size, and needed power to operate.
When a Backlight goes bad, it just goes all at once. Not goes dark as you have stated.
(Your statement meant to me that you power on the laptop, the LCD screen shows well, then goes dark)
How to install an Inverter in a Toshiba Satellite P200 series laptop,http://www.irisvista.com/tech/laptops/toshiba-satellite-p205/take-apart-display-panel-1.htm
(You can click on each individual photo to enlarge. You are on Page 1. Page 2 is accessed at the bottom, by clicking on it. Page 1::Page 2::Find another model)
Just one of many places that offers an Inverter for your Satellite P200.http://www.notebooksolutions.ca/zc/index.php?main_page=product_info&products_id=5367&zenid=43bc1e775a9ed2d4309140111231cab5
1) ALL power removed from the laptop before beginning.
AC adapter (Charger), and Battery.
2) The screws you will remove, may have screws that look alike, but are NOT.
Do Not mix them up.
For this reason it is suggested to use a multi-compartment container, and label the compartments for where the screws came out of.
You can use a clean egg container, and label the individual egg holders, in a pinch.
IMHO it isn't hard to perform this repair, and you can do it. Save yourself money.
For additional questions regarding this, or comments, please post in a Comment.