  When selecting flux-limited cluster samples, the detection efficiency of
X-ray instruments is not the same for centrally-peaked and flat objects, which
introduces a bias in flux-limited cluster samples. We quantify this effect in
the case of a well-known cluster sample, HIFLUGCS. We simulate a population of
X-ray clusters with various surface-brightness profiles, and use the
instrumental characteristics of the ROSAT All-Sky Survey (RASS) to select
flux-limited samples similar to the HIFLUGCS sample and predict the expected
bias. For comparison, we also estimate observationally the bias in the HIFLUGCS
sample using XMM-Newton and ROSAT data. We find that the selection of X-ray
cluster samples is significantly biased ($\sim29%$) in favor of the peaked,
Cool-Core (CC) objects, with respect to Non-Cool-Core (NCC) systems.
Interestingly, we find that the bias affects the low-mass, nearby objects
(groups, poor clusters) much more than the more luminous objects (i.e massive
clusters). We also note a moderate increase of the bias for the more distant
systems. Observationally, we propose to select the objects according to their
flux in a well-defined physical range excluding the cores,
$0.2r_{500}-r_{500}$, to get rid of the bias. From the fluxes in this range, we
reject 13 clusters out of the 64 in the HIFLUGCS sample, none of which appears
to be NCC. As a result, we estimate that less than half (35-37%) of the galaxy
clusters in the local Universe are strong CC. In the paradigm where the CC
objects trace relaxed clusters as opposed to unrelaxed, merging objects, this
implies that to the present day the majority of the objects are not in a
relaxed state. From this result, we estimate a rate of heating events of
$\sim1/3$ Gyr$^{-1}$ per dark-matter halo.
