General GPS module question

I'm completely new to Arduino and GPS modules so I have a few questions I'd like to ask before I commit to purchasing items.

If I had a pair of Aduinos with individual GPS modules attached and placed them within a mile or two of each other, would the received GPS time on both of these devices be identical (or nearly identical to within a +/- 1sec)?

I'm trying to design a mountain biking race timing system and I'm trying to determine the best way I can synchronize time stamps between a start and finish station for accurate timing. The only other solution I thought of would be to wirelessly network the two stations using Xbees so that the finish station can transmit back to the start station when the racer crosses the finish line. However, this poses a problem due to topography inherent to mountain biking. Wireless signals may not be reliable and repeaters may be needed.

The alternative I thought of was to use GPS modules to basically have a synchronized clock between the start and finish stations. This way when a racer crosses the start and finish stations, the time stamps are in the same time frame of reference.

The time should be the same well within one second. If done just right you can probably get them synchronized to a few microseconds.

Fantastic news. I will try making this timing system using GPS modules then. :)

Thanks for your help! I probably will be back with other questions invariably since I haven't worked with microcontrollers in about a decade!