  Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has
cardinality $n$. Consider linear extensions of $P$ as permutations
$x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we
define $\mathbb{P}(x\prec y)$ to be the proportion of linear extensions of $P$
in which $x$ comes before $y$. For $0\leq \alpha \leq \frac{1}{2}$, we say
$(x,y)$ is an $\alpha$-balanced pair if $\alpha \leq \mathbb{P}(x\prec y) \leq
1-\alpha.$ The $1/3-2/3$ Conjecture states that every finite partially ordered
set which is not a chain has a $1/3$-balanced pair. We make progress on this
conjecture by showing that it holds for certain families of posets. These
include lattices such as the Boolean, set partition, and subspace lattices;
partial orders that arise from a Young diagram; and some partial orders of
dimension $2$. We also consider various posets which satisfy the stronger
condition of having a $1/2$-balanced pair. For example, this happens when the
poset has an automorphism with a cycle of length $2$. Various questions for
future research are posed.
