The detection of planar structures within the satellite systems of both the Milky Way (MW) and Andromeda (M31) has been reported as being in stark contradiction to the predictions of the standard cosmological model ($Lambda$CDM). Given the ambiguity in defining a planar configuration, it is unclear how to interpret the low incidence of the MW and M31 planes in $Lambda$CDM. We investigate the prevalence of satellite planes around galactic mass haloes identified in high resolution cosmological simulations. We find that planar structures are very common, and that ~10% of $Lambda$CDM haloes have even more prominent planes than those present in the Local Group. While ubiquitous, the planes of satellite galaxies show a large diversity in their properties. This precludes using one or two systems as small scale probes of cosmology, since a large sample of satellite systems is needed to obtain a good measure of the object-to-object variation. This very diversity has been misinterpreted as a discrepancy between the satellite planes observed in the Local Group and $Lambda$CDM predictions. In fact, ~10% of $Lambda$CDM galactic haloes have planes of satellites that are as infrequent as the MW and M31 planes. The look-elsewhere effect plays an important role in assessing the detection significance of satellite planes and accounting for it leads to overestimating the significance level by a factor of 30 and 100 for the MW and M31 systems, respectively.
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