In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that
$x$
and
$y$
commute in a compact group’, Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group
$G$
the probability
$d(G)$
that two randomly selected elements
$x, y\in G$
satisfy
$xy=yx$
, and we discussed the remarkable consequences on the structure of
$G$
which follow from the assumption that
$d(G)$
is positive. In this note we consider two natural numbers
$m$
and
$n$
and the probability
$d_{m,n}(G)$
that for two randomly selected elements
$x, y\in G$
the relation
$x^my^n=y^nx^m$
holds. The situation is more complicated whenever
$n,m\gt 1$
. If
$G$
is a compact Lie group and if its identity component
$G_0$
is abelian, then it follows readily that
$d_{m,n}(G)$
is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group
$G$
: for any nonopen closed subgroup
$H$
of
$G$
, the sets
$\{g\in G: g^k\in H\}$
for both
$k=m$
and
$k=n$
have Haar measure
$0$
. Indeed, we show that if a compact group
$G$
satisfies this condition and if
$d_{m,n}(G)\gt 0$
, then the identity component of
$G$
is abelian.