Probability Is Exact Only in Theory: How Real-World Variables Break the Model

Probability theory is exact as mathematics, but its statements about the world are conditional on assumptions about the mechanism, independence, stability, the set of outcomes, and who knows and does what. This preprint reviews documented violations in coins, dice, cards, roulette and finance, then tests the idea on the 100-prisoners problem, whose optimal loop strategy has an exact survival probability of 31.18%. With every random choice drawn from atmospheric radio noise (random.org), the loop strategy survived 133 of 400 games (33.3%; 95% interval 28.8–38.0%), and 31.19% of 1,000,000 seeded simulated games; the random strategy never survived. Three sources of randomness, from a deterministic formula to radio noise, were indistinguishable. Controlled breaks of the model's assumptions were not: fewer than 50 random swaps from sorted order guarantee survival; a hostile warden who builds one long loop drives it to zero; if 1% of prisoners abandon the plan, survival falls to 18.9%; and when prisoners can see each other's results, random search matches the loop strategy and survives the hostile warden. One imperfect riffle shuffle changed nothing. All implementations were checked against exhaustive enumeration of small cases. Probability does not fail at random; it fails where its assumptions fail. Supplementary files: prisoners_experiment.py (Python 3, standard library only) reproduces every table and figure and runs the exact small-case checks. atmospheric_run1.txt and atmospheric_run2.txt are the raw random.org bytes; running "python3 prisoners_experiment.py replay atmospheric_run1.txt 100" and "python3 prisoners_experiment.py replay atmospheric_run2.txt 300" reproduces the physical-randomness results exactly.