The original

At least, the first time I heard it... told to me by the son of an engineer, then studying mathematics as was I... (Apologies for the non-inclusive language, this is as I heard it.)

Three people are given the problem:

Prove, by the means of inference regarded as valid by your profession, that all odd numbers are prime.

The chemist goes first. He reasons "Three is prime, five is prime, seven is prime... this is easy... ergo, all odd numbers are prime."

The physicist goes next. He reasons "Three is prime, five is prime, seven is prime, nine... nine is not prime, but eleven and thirteen are, so we conclude that nine is experimental error, ergo all odd numbers are prime."

Last goes the engineer. He reasons "Three is prime, five is prime, seven is prime, and so are nine, eleven, thirteen, fifteen, seventeen, nineteen and twenty-one. Ergo all odd numbers are prime."

Some more good ones

There have been many variations published on the Internet, but two particularly appeal to me. The phrasing below is my own.

Next comes an architect. They observe: "Three is prime, five is prime, seven is prime, nine... we'll have to come back to that I think... eleven is prime, thirteen is prime... I'm sure my engineer can work something out for nine... "

Finally (for now) a lawyer: "Three is prime, five is prime, seven is prime... there may be a prima facie case that nine may possibly not be prime, but based on the overwhelming, in point of fact the unanimous weight of precedent as we have just observed, it is strongly indicated that nine may and even should be deemed to be prime."

There are many more, including many from computing which are funny only to those intimately familiar with particular aspects of it, and many which are just variations of the methodology of the chemist or physicist, and/or again only funny to those who know a particular jargon. So that will do for now!