One of the reasons we don't all speak German today is the combination of elementary group theory, and universal Turing machines.
Per a request below, I'll elaborate: The British, French and Poles had a joint intelligence effort against Germany during the 30s (at that time breaking cyphers was considered a liberal-arts high classes hobby). The French, using traditional cloak and dagger techniques, managed to get their hands on the German cypher machine (Enigma) plans - but did not know what to do with it. The Polish cypher bureau however had an amazing idea - they'll let math students play with it. From a group theoretic point of view, the machine worked by taking products in S_{letters}. However, the machine designers did not notice that the products were all of the form A^{-1} R A - (where A is constantly changing but R is not) which enabled the poles to discover the cycle structure of R. You can look here for more details.
When the Germans invaded Poland, the Poles let the French and the British on the secret, and the British took over the operation. The Poles used machines (bombes) from the very start to break the codes, but as the Germans improved the coding machines, the need arose the reconfigure the breaking machines as they run. Luckily for everyone involved, the head cryptanalysis on the British side was Turing, who built such a machine: Colossus.
Finally, a funny bit which I hear in Bletchley park last year: "everybody" knows that at the end of the war, Menzies and Churchill ordered the Collosi destroyed and the plans burnt, and yet they were rebuilt a few years after they were declassified - how come ?
The easy part is that some of the engineers saw it coming and took backup plans home. However, it is much funnier to discover that the Royal post office continued to manufacture the parts until the 70s, and they were still in stock.
Perhaps Euler's polyhedral formula:
V(vertices) + F(faces) - E(edges) = 2
provides an example of what you mean?
Euler did not give a proper proof but shortly thereafter this result inspired huge advances that had dramatic effects on the evolution of geometry, topology, convexity, and what today is called graph theory.
One measure of how rich this topic is can be seen from the many types and styles of proofs that can be found for this result collected below by David Eppstein:
http://www.ics.uci.edu/~eppstein/junkyard/euler/
Best Answer
A tragic example of this is the case People v. Collins, in which a prosecutor asked a mathematician (as an expert witness) a question of the form, "assuming these events are independent, what is the probability that...". The events were obviously not independent, things like "drives a convertible", "has a caucasian girlfriend", "girlfriend has blond hair", and some others. The mathematician answered the misleading question correctly (assuming independence), and the defendant went to jail. The California Supreme Court later overturned the verdict, in a decision that shows a surprisingly solid understanding of probability.
This case could be required reading (the supreme court decision, anyway) in any introduction to probability course. It has counting, independence, and conditional probability all involved in a fundamental way.