# Basic examples of probabilistic method

I'm looking for a truly basic example of probabilistic method proof which could be presented without a board (i.e. speaking only), that is, even moderately complicated calculations are not allowed.

That said, it would be best if the problem was self-contained, e.g. I've searched the web a lot, but most of these come from graph theory where you have to explain first what a graph is.

It does not matter if the result can be easily proved by some other means (but it would be great if the statement wasn't blatantly true at the first glance).

What I came at is

Let $A_1, A_2, \ldots, A_n$ be some finite sets. Then we can pick some of them such that at least half of the underlying elements are repeated odd number of times (i.e. belong to the xor of the picked sets).

but it feels a bit rough. Any better suggestions?

## 1 Answer

How about a proof of the basic fact that the sum of angles of a triangle (in Euclidean space) is 180 degrees? (Is it "blatantly true"?)

Proof: Let the interior angles be $$a, b$$ and $$c$$. Pick a direction uniformly at random and project the triangle along this direction. The expected number of disappearing vertices (that is, the ones that do not appear as vertices in the shadow segment) is precisely one. Also the individual vertices disappear with probabilities $${2a\over 360}, {2b\over 360}, {2c\over 360}$$. By linearity of expectation $$a+b+c=180$$.

• I don't understand, you are projecting triangle to what? Some (fixed) line? Commented Jun 13, 2020 at 12:27
• @Aqua Fix any line perpendicular to the randomly chosen direction. Project the triangle onto that line, forming a "shadow" interval. The probability that a particular vertex is projected into the interior of the shadow (i.e., "disappears") is just twice the proportion of $360^\circ$ that's occupied by its interior angle (in degrees) -- twice, because projecting in any particular direction gives the same result as projecting in the opposite direction. Commented Aug 23, 2020 at 17:27
• Thanks! @r.e.s. Commented Aug 23, 2020 at 17:29