Blazars, a subclass of active galactic nuclei, 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 γ-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, especially during flares where time-dependent neutrino searches may have a higher detection probability. 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, wederive the mean event rate above 100 TeV (∼0.57 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 γ-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 ± 0.60 (2.73 ± 0.38) νμ +νbarμ 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 γ-rays and neutrinos is assumed, predicts 1.60 ± 0.16νμ +νbarμ events. We conclude that a non-detection of high-energy neutrinos by IceCube would probe the neutrino/γ-ray flux correlation during major flares or/and the hadronic contribution to the blazar emission.