I think the fact that e^i*PI +1 = 0 surprises almost everyone when they
first hear of it. I was surprised to learn that infinity times infinity is
just the same old infinity but 2 to the power of infinity yields a larger
infinity, and I was surprised to learn that there is a proof that some
things are true but a proof that they are true will never be found because
it does not exist.

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Probability gave me a lot of surprises, in grade school I was surprised to
learn that in a group of just 23 people the chance that two of them have
the same birthday is 50% and with 57 people it's 99%; and it took me an
embarrassingly long time to understand the Monty Hall puzzle, the idea that
if you change your guess when Monty gives you the opportunity your chance
of winning the car behind the door increases from 1 chance in 3 to 2
chances in 3.
John K Clark
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