The probability to observe a specific number of strange and multi-strange hadrons ($n_s$), denoted as $P(n_s)$, is measured by ALICE at midrapidity ($|y|<~0.5$) in $\sqrt{s} = 5.02$ TeV proton-proton (pp) collisions, dividing events into several multiplicity-density classes. Exploiting a novel technique based on counting the number of strange-particle candidates event-by-event, this measurement allows one to extend the study of strangeness production beyond the mean of the distribution. This constitutes a new test bench for production mechanisms, probing events with a large imbalance between strange and non-strange content. The analysis of a large-statistics data sample makes it possible to extract $P(n_s)$ up to a maximum $n_s$ of 7 for K$^{0}_{\rm s}$, 5 for $Λ$ and $\barΛ$, 4 for $Ξ^-$ and $Ξ^+$, and 2 for $Ω^-$ and $Ω^+$. From this, the probability of producing strange hadron multiplets per event is calculated, thereby enabling the extension of the study of strangeness enhancement to extreme situations where several strange quarks hadronize in a single event at midrapidity. Moreover, comparing hadron combinations with different $\it{u}$ and $\it{d}$ quark compositions and equal overall $s$ quark content, the contribution to the enhancement pattern coming from non-strangeness related mechanisms is isolated. The results are compared with state-of-the-art phenomenological models implemented in commonly used Monte Carlo event generators, including PYTHIA 8 Monash 2013, PYTHIA 8 with QCD-based Color Reconnection and Rope Hadronization (QCD-CR + Ropes), and EPOS LHC, which incorporates both partonic interactions and hydrodynamic evolution. These comparisons show that the new approach dramatically enhances the sensitivity to the different underlying physics mechanisms modeled by each generator.