I love it when they say 50% chance of rain. 50% - that's a coin flip, right? That's not science!
No, that is not what that means.
In a nutshell it is a calculation of coverage of precipitation over a pre-defined area times confidence. So your simplified "coin flip" scenario is only valid when there is exactly 50% coverage of precipitation and the forecaster is 100% confident. Neither rarely happens. Meteorology is not an exact science...there is probably still far more we still don't know about how the atmosphere works than we understand.
FAQ from the NWS:
What does this "40 percent" mean? ...will it rain 40 percent of of the time? ...will it rain over 40 percent of the area?
The "Probability of Precipitation" (PoP) describes the chance of precipitation occurring at any point you select in the area.
How do forecasters arrive at this value?
Mathematically, PoP is defined as follows:
PoP = C x A where "C" = the confidence that precipitation will occur somewhere in the forecast area, and where "A" = the percent of the area that will receive measurable precipitation, if it occurs at all.
So... in the case of the forecast above, if the forecaster knows precipitation is sure to occur ( confidence is 100% ), he/she is expressing how much of the area will receive measurable rain. ( PoP = "C" x "A" or "1" times ".4" which equals .4 or 40%.)
But, most of the time, the forecaster is expressing a combination of degree of confidence and areal coverage. If the forecaster is only 50% sure that precipitation will occur, and expects that, if it does occur, it will produce measurable rain over about 80 percent of the area, the PoP (chance of rain) is 40%. ( PoP = .5 x .8 which equals .4 or 40%. )
In either event, the correct way to interpret the forecast is: there is a 40 percent chance that rain will occur at any given point in the area.
http://www.srh.noaa.gov/ffc/?n=pop
...so now you know.
By the way, Doctors and lawyers dogcatchers policemen...no one is perfect. Some jobs allow for or require educated guesses, and informed decisions.