Here is one example of what is mathematics as opposed to "layman reasoning" that I was told about by Yuval Peres. During WWII, the British airplanes were often shut down by Germans because the armor was too weak. The decision was made to put some extra armor on the planes but, since you can add only that much weight to a plane without affecting its performance, only a few selected areas could be enforced. Upon looking at the damaged planes that returned to the base, the engineering committee recommended to put armor to the most frequently damaged areas, as seen on the planes available for investigation. It took a statistician to explain why you should do exactly the opposite.

Here is one example of what is mathematics as opposed to "layman reasoning" that I was told about by Yuval Peres. During WWII, the British airplanes were often shot down by Germans because the armor was too weak. The decision was made to put some extra armor on the planes but, since you can add only that much weight to a plane without affecting its performance, only a few selected areas could be enforced. Upon looking at the damaged planes that returned to the base, the engineering committee recommended to put armor to the most frequently damaged areas, as seen on the planes available for investigation. It took a statistician to explain why you should do exactly the opposite.