What do weather forecasters mean when they say "50% chance of rain"? Even more confusing: weather report often says something like "30% chance of rain. >10mm", then the next day "70% chance of rain <1mm". How should I deal with this information, for example, if I'm planning a picnic?
In the US, meteorologists forecast the probability of ANY amount of precipitation falling. The minimum amount of that we deem acceptable to meet this criteria is .01". So, we are forecasting the probability of one hundredth of an inch of precipitation to fall at a location.
We look at observational data from ground stations, satellites, and computer models and combine it with our knowledge of meteorology/climate and our past experiences with similar weather systems to create a probability of precipitation.
A 10% chance of rain means that 10 times out of 100 with this weather pattern, we can expect at least .01" at a given location.
Likewise, a 90% chance of rain would mean that 90 times out of 100 with this weather pattern, we can expect at least .01" at a given location.
Similarly, the Storm Prediction Center issues severe threat probabilities. You can see things like 5% chance of tornado or 15% chance of severe hail.
This simply means 5 times out of 100 in this scenario we can expect a tornado. Or 15 times out of 100 you can expect severe hail with thunderstorms.
By definition, there is no difference in the amount of rain forecast by a 10% chance or a 90% chance. Instead, that information is defined elsewhere, typically by a Quantitative Precipitation Forecast (QPF).
Meteorologists are still struggling with the best ways to inform the public about the differences between high chance, low QPF events versus low chance, high QPF events, and everything else in between.
Meteorologists are still human and have their own wet or dry biases that can hedge the chance of precipitation you see. Recently, forecasts are relying more on bias-correction techniques and statistical models to remove the human bias from precipitation forecasts.
To build on Drew's excellent answer, I would note the following:
Weather forecasts of the future are built on historical frequencies. A "forecast" of a X percent chance of rain (however defined), means, based on the current configuration of casual variables (temperature, humidity, air pressure etc.), there has historically been an X per cent chance of rain in 24 hours. Therefore, we forecast X percent for tomorrow.
Sometimes a forecaster will try to incorporate unusual events into the equation. (Chernobyl comes to mind.) Then the argument might be, historically the chances of rain were X percent, but we're bumping up the forecast to y percent, because of all the atomic particles released by Chernobyl.
A 50% chance of rain means that there is a 50% chance that at least some rain will fall at any given location within the area covered by the weather forecast, within the time period covered by the weather forecast period. It does not mean that it will rain 50% of the time.