If a popular website provides an hourly forecast, and it states at 12:00pm there will be a 20% chance of precipitation and 2.5 mm of rain...
What does the "2.5 mm of rain" mean?
(A) ... If it does rain, on average there will be 2.5 mm of rain.
( 20% * 2.5mm + 80% * 0.0mm ) = 0.5 mm average rainfall
(B) ... "2.5 mm of rain" is a "statistical prediction" which means that a huge storm could be coming and that if it does rain there will be about ~12.5 mm of rain for the hour but if the storm misses then there will be 0.0 mm of rain.
( 20% * 12.5mm + 80% * 0.0mm ) = 2.5 mm average rainfall
How should a person interpret a weather forecast of 2.5 mm with a 20% probability? There is a big difference between potentially 2.5 mm of rain and 12.5 mm of heavy rain.
In (A), a typical person reading may imagine how deep 2.5 mm is and not be worried as that is just a thin film of water. However if a site reported 12.5 mm, they may imagine puddles and hence be worried. Since a typical person is not mathematically inclined and unlikely to be knowledgeable in conditional probabilities, using (A) is potentially more useful for most people.
In (B), a typical person reading imagines how deep 2.5 mm is and is not worried. That person will be very surprised if the storm hits. A statistical prediction is useful for science/math people and keeps the prediction independent of the PoP %. It is however very misleading for 95-99% of people that happen to be not versed in meteorology. I assume (B) is used but wanted to confirm.