Given is a positive integer n. What fraction of the non-empty subsets of $\{1,2, \ldots, 2 n\}$ has an odd smallest element?

(Birgit Vera Schmidt)
For each $k$ with $1 \leq k \leq 2 n$, the number of subsets that have $k$ as their smallest element is $2^{2 n-k}$. (Each element greater than $k$ can either be included or not included.)

The number $U$ of subsets with an odd smallest element is thus

$$
U=2^{2 n-1}+2^{2 n-3}+\cdots+2^{3}+2^{1}=2 \cdot\left(4^{n-1}+4^{n-2}+\cdots+4^{1}+4^{0}\right) .
$$

The number $G$ of subsets with an even smallest element is

$$
G=2^{2 n-2}+2^{2 n-4}+\cdots+2^{2}+2^{0}=4^{n-1}+4^{n-2}+\cdots+4^{1}+4^{0} .
$$

It follows that $U=2 G$ and the desired ratio is $2 / 3$.

(Birgit Vera Schmidt)