The probability is temporal as well as spatial.
"The probability of precipitation has the following features:
The likelihood of occurrence of precipitation is stated as a percentage...
The probability is for a specified time period...
The probability forecast is for any given point in the forecast area...
"Here are two examples giving the same result:
(1) If the forecaster was 80% certain that rain would develop but only expected to cover 50% of the forecast area, then the forecast would read "a 40% chance of rain" for any given location.
(2) If the forecaster expected a widespread area of precipitation with 100% coverage to approach, but he/she was only 40% certain that it would reach the forecast area, this would, as well, result in a "40% chance of rain" at any given location in the forecast area."
http://www.utexas.edu/depts/grg/kimmel/nwsforecasts.html