Yippee! All you said is that "if there's an 80% chance it'll rain, they mean there's a 20% chance it won't rain." I bow to your mathematical prowess. Okay, so let's say there's a 1 in 5 chance the storm system could change course - yet out of the last 500 times there's been a "1 in 5 chance", the storm hasn't changed. The person determining that 1 in 5 chance seems to be getting that determination wrong.
I understand this: "So if there’s a 50% chance that it will rain somewhere in the area, and, if it does rain, 80% of the area is likely to be covered, the probability of precipitation is 40%," and the rest of the point you thought you were making. Let's say that's the forecast 2000 days in a row. Then, 50% of the time, it'll rain "somewhere in the area." And, when it does rain, "80% of the area is likely to be covered." If you were somewhere in the area, that works out to... drum roll... 40% of the time that they say "there's a 40% chance of rain", it rains in any particular area.
Unless that is, you're saying that I'm an unlucky bastard, and EVERY time there's an 80% chance of rain, it rains where I'm at. And, somewhere, maybe 20 miles away, maybe 50 miles away, there's some really lucky bastard who, when they say "80% chance of rain" it doesn't always rain.