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 ".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%. )
If you look at an hourly forecast and see three hours of 10%, followed by an hour at 20% and an hour at 50%, that doesn't mean that there's now a 100% chance of rain that day, it means that it's going to get cloudier as the day goes and and the chance in that last hour is still 50%. [...]
To compare it to coin flips, though, you wouldn't be looking for the outcome of individual coin flips. Each individual coin flip does carry a 50/50, heads or tails probability, but looking for the same outcome two times in a row (heads then heads again) has a probability of 25%, [...]
To compare it to coinflips the number of outcomes increases as their chance drops to 25%. Heads or Tails (H or T) becomes ( HH or HT or TH or TT).
Thanks for looking up the PoP definition. It is curious how they apply a probability to an area. But it doesn't change anything mathematically. Now that I have opportunity to procrastinate, I thought about it and basically A (area%) is just another probability. It goes from 0 to 100%. Whereas C (confidence) is more or less = 1 in the actual weather simulation, which by the way are extremely accurate and reliable about 3 days ahead.
Rain isn't usually a random event, cloud density and low air pressure means very high certainty of rain. However selecting a time and place where it rains is like a random experiment. If a thick cloud covers half of an area, but you don't know which half will get wet, then it is exactly the same as if thin clouds (50% confidence) covered the entire area.
50% PoP means you are going to get wet an average 50 out of a 100 times with this same forecast, even though your umbrella just occupies one square meter or 9 sq. ft.
Combining predictions for large areas doesn't make much sense, it always rains somewhere. However combining time intervals makes sense for as long as you have to plan ahead.
- 3 hours of 10%, 1 hour of 20% and 1 hour of 50% would "add" up to a (1-0.9³ * 0.8 * 0.5) = ~71% PoP for 6 hours ahead.