My forecast app is saying "0% chance of precipitation, 50% chance of thunderstorms".

?

Probability of Precipitation is actually two items: confidence and area. Probability = C x A. So if a forecaster is 80% certain that rain will fall, but it will only affect 20% of the area, then probability of rain =

.8 x .2 = .16 or 16% chance.

So in this case, the confidence of rain and the area of rain is rather low. Nevertheless, if it occurs, then there is a 50% chance it will be a thunderstorm, which makes sense for IL and those sudden popup afternoon thunderstorms.

Perhaps a C = 10% and an A = 1%. Probability of rain is .1 x .01. Or 0.1%, which would be given as 0%.

Or as JeffDG has pointed out, in some cases, a thunderstorm will occur with no precipitation as it evaporates before hitting the ground. This is known as a "Dry Thunderstorm". Generally these are common in the American West, but can happen in other places. The threshold for a dry thunderstorm is 0.1 inches of rain.

Or what will also happen is the information is updated at different times and you just "get lucky" by accessing the data before it has been updated. For example, I looked at the weather and it stated a 10% chance of rain, but it was already raining for my area. It took about 15-30 minutes to get updated to a 100% chance.