Luck is widely invoked yet rarely formalized. We propose a probabilistic framework that reframes luck and unluck as measurable deviations from random expectation, introducing a single summary statistic-the Unluck Coefficient (UC)-that captures the expected magnitude of downside deviation in any allocation system. The framework rests on three ingredients: a min-max normalized burden score s, a theoretically derived baseline τ , and a signed luck deviation ∆ whose sign partitions outcomes into "lucky" and "unlucky" classes. We validate the framework across five progressively complex settings: (i) fair coin flips, (ii) random digit arrays (10 positions, 10 5 trials), (iii) fair-die (Ludo) six-counts, (iv) a birthday-style random-to-bin allocation, and (v) a large-scale pixel study over 1.86 M random 7×7 image patches (49 positions each). In the multi-position symmetric settings (coins, arrays, birthday bins), the UC clusters near the half-normal reference 1/ √ 2π ≈ 0.40 1 as sample size grows. In the pixel study, significant non-uniformity persists: a chisquare statistic of 5 249.71 (df = 48, p ≪ 10-6), nine positions significant after Benjamini-Hochberg FDR control at q = 0.05, and a positive net luck surplus i ∆ i = 8.19. We further derive resource-planning curves showing that the UC decays as c/ √ R, formalizing the intuition that "bad luck" can be mitigated but never eliminated without unbounded resources. By grounding folk notions of fortune and misfortune in standard probability theory, this work offers a rigorous yet accessible bridge between the philosophy of luck, the statistics of random allocation, and practical downside-risk assessment.
Md Takrim Ul Alam (Thu,) studied this question.