Today March 28 there was blue sky this morning and bright sunshine. Then around 11:30 the sky turned black and there was steady rain thunder and lighting. The world looked like it was going to end. It rained everywhere for miles around and there was a little wind. But, after two hours the sun came out and everything was just as it was before.
So in fact it did indeed rain, with thunderstorms in the afternoon, as the weather forecast stated. What's more the forecaster gave a PoP of 60%, meaning there was a 60% chance of precipitation occurring at any particular location in the forecast area, so fairly accurate.
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 measureable 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 ".6" which equals .6 or 60%.)
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%. )