  Blazars are prime candidate sources for the high energy neutrinos recently
detected by IceCube. Being one of the brightest sources in the extragalactic
X-ray and $\gamma$-ray sky as well as one of the nearest blazars to Earth, Mrk
421 is an excellent source for testing the scenario of the blazar-neutrino
connection. Here, we model the spectral energy distribution of Mrk 421 during a
13-day flare in 2010 with unprecedented multi-wavelength coverage, and
calculate the respective neutrino flux. We find a correlation between the $>1$
PeV neutrino and photon fluxes, in all energy bands. Using typical IceCube
through-going muon event samples with good angular resolution and high
statistics, we derive the mean event rate above 100 TeV ($\sim0.57$ evt/yr) and
show that it is comparable to that expected from a four-month quiescent period
in 2009. Due to the short duration of the flare, an accumulation of similar
flares over several years would be necessary to produce a meaningful signal for
IceCube. To better assess this, we apply the correlation between the neutrino
and $\gamma$-ray fluxes to the 6.9 yr Fermi-LAT light curve of Mrk 421. We find
that the mean event count above 1 PeV for the full IceCube detector livetime is
$3.59\pm0.60$ ($2.73\pm0.38$) $\nu_\mu+\bar{\nu}_\mu$ with (without) major
flares included in our analysis. This estimate exceeds, within the
uncertainties, the $95\%$ ($90\%$) threshold value for the detection of one or
more muon (anti-)neutrinos. Meanwhile, the most conservative scenario, where no
correlation of $\gamma$-rays and neutrinos is assumed, predicts $1.60\pm0.16$
$\nu_\mu+\bar{\nu}_\mu$ events. We conclude that a non-detection of high-energy
neutrinos by IceCube would probe the neutrino/$\gamma$-ray flux correlation
during major flares or/and the hadronic contribution to the blazar emission.
