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You know that no matter how many examples you give to show that a conjecture is true, it does not prove the conjecture. It takes just one counterexample to disprove a conjecture. Describe a conjecture you thought was true based on many examples, but that turned out to be false based on a counterexample. The conjecture does not have to be mathematical.
Great question! The underlying principle is showing the difference between deductive and inductive reasoning. One interesting example would be to say what people do when someone yells "Fire" in a crowded theater. You might observe everyone panicking. And you might be in the same situation over and over and always see everyone panicking, and then conclude that everyone panics in that situation, until you see the first person who doesn't and that would shoot down your conjecture.
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That would be an example of deductive reasoning because you would be deducing (even though incorrectly) that everyone panics. You see lots of isolated examples and are then trying to come up with a general principle.