Why does the probabilistic method work to obtain bounds?

Question

In the following problem, the probabilistic method is used to obtain a lower bound.

$15$ chipmunks live in $52$ circularly arranged trees. Show there must be a neighborhood of $7$ trees that houses at least three chipmunks.
Let $X$ denote the number of chipmunks living in the neighbohood of a randomly chosen tree. Let $X_i$ equal $1$ iff chipmunk $i$ lives in the neighborhood of said tree. Then $$\mathbb{E}(X)=\sum_{i=1}^{15}\mathbb{E}(X_i)=\frac {15\cdot 7}{52} > 2 $$ My question is: why does this work?
I believe that here, we are trying to obtain the minimum number achievable, in other words, we are trying to obtain the lower bound for $\min f(x)$ where $f$ counts the number of chipmunks living in the neighborhood of a tree. Yet, I do not see how the expected value serves this purpose. I can see how the expected value can bound the maximum from below and the minimum from above, but I cannot see how to bound them from above and below, respectively.

Answer 1

If we have a discrete variable $Y$ and we know that $E(Y)>a$, then we automatically know that $P(Y > a)>0$. That is, there exists some value $b>a$ with positive probability: $P(Y=b)>0$

In your case: $E(X)>2 \implies P(X>2)>0$, then there exists some $b \in \{3, 4 \cdots \} $ with $P(X=b)>0$, then there exists "a neighborhood of $7$ trees that houses at least three chipmunks".