Proof that doctors could relate to [closed]

Solution 1:

My go-to example of this type is the following problem: It's easy to see how to cover a chessboard with 32 dominoes, each of which covers two adjacent squares. Suppose the chessboard has two diagonally opposite corners deleted. Is it possible to cover the remaining 62 squares with 31 dominoes?

If you try it, you quickly find that it is much more difficult than covering the full chessboard, although you might not be convinced that it was impossible. But it isn't possible. Each domino covers one black and one white square, so the 31 dominoes must cover 31 black and 31 white squares. But the mutilated chessboard has 32 squares of one color and only 30 of the other color.

I think anyone can understand this—and I have presented it to administrative assistants with no mathematical training who understood it right away—but I think it perfectly captures the essential feature of mathematics, which is that we don't consider all the possible ways of placing the dominoes one at a time, but instead we find an underlying regularity of structure and show that the solution, if it existed, would require a different kind of structure.

Solution 2:

Bayes' theorem!

If a test for a disease is $99$% accurate, and $1$% of the population has a disease, what is the probability that someone who tests positive for the disease actually has the disease?