  In phylogenetic studies, biologists often wish to estimate the ancestral
discrete character state at an interior vertex $v$ of an evolutionary tree $T$
from the states that are observed at the leaves of the tree. A simple and fast
estimation method --- maximum parsimony --- takes the ancestral state at $v$ to
be any state that minimises the number of state changes in $T$ required to
explain its evolution on $T$. In this paper, we investigate the reconstruction
accuracy of this estimation method further, under a simple symmetric model of
state change, and obtain a number of new results, both for 2-state characters,
and $r$--state characters ($r>2$). Our results rely on establishing new
identities and inequalities, based on a coupling argument that involves a
simpler `coin toss' approach to ancestral state reconstruction.
