A law can determine a distribution without fixing the next individual outcome. Probability introduces a structured account of alternatives; it does not remove the need for dynamics, preparation and empirical checks.
A lawful distribution of unpredictable steps
Take a particle that moves one unit right or left at each tick. Let successive steps be independent, with rightward probability p. After n ticks its position is the sum of n signed steps. The expected position is n(2p − 1), while the variance is 4np(1 − p). These follow by adding the means and variances of independent increments.
For n = 100 and p = 0.6, the mean displacement is 20 and the standard deviation is √96, approximately 9.80. The mean is not a trajectory that every particle follows. It summarizes an ensemble containing many paths, each built from definite left or right steps.
Independence does substantial work. If all 100 steps instead copy one shared initial sign, the mean remains 20 but the variance becomes 9,600. The same one-step probabilities can coexist with radically different long-run behaviour. A probabilistic law needs joint structure across time, not only a list of probabilities for isolated events.
Random timing can have a precise rule
Suppose a particular event has a constant rate λ per unit time, with no ageing of its waiting-time distribution. The survival probability is S(t) = exp(−λt), the probability that the event has not occurred by t. With λ = 2 per second, survival at half a second is exp(−1), approximately 0.368. The mean waiting time is half a second.
The assumption also gives S(t+s)/S(t) = S(s). Having waited t without an event does not change the remaining waiting-time law. A device whose failure becomes more likely as it wears out generally requires a different hazard model. Calling both timings “random” hides that important distinction.
Gillespie's stochastic reaction method is an established example of calculating event times and event choices from specified reaction propensities. Its exactness concerns the chosen stochastic model and its assumptions; it does not make every chemical approximation exact. The exponential example above is an independently worked single-rate illustration.
Several reasons for using probability
One model uses probability because the preparation is incompletely known. Another omits microscopic variables and describes their aggregate influence statistically. A third treats probabilistic outcomes as part of its fundamental rule. Those interpretations can agree on a limited set of observable frequencies.
A deterministic generator with a random initial seed illustrates the first distinction. Conditional on the seed, every subsequent output may be fixed. Across the preparation ensemble, outputs have a distribution. The reader of the output stream may be unable to distinguish it from another mechanism over the available observations. Such agreement does not prove that all physical randomness is ignorance of a seed.
Quantum theory supplies probabilities through its state and measurement rules. Whether a proposed deeper account reproduces them is a separate calculation, especially for entangled systems and different measurement settings. A classical random walk is useful for understanding variance; it is not a derivation of the Born rule.
Test the distribution and its dependencies
A model that correctly predicts an average can still fail badly. For the two walks above, checking only mean displacement misses the hundredfold variance difference. One should also inspect fluctuations, correlations, waiting times and responses to changed preparations. A narrow successful test supports only the aspects it probes.
For the independent walk, repeated experiments permit a distribution of sample means to be calculated. That uncertainty belongs to the experimental estimate, whereas the spread of individual paths belongs to the modelled process. Confusing these two spreads can turn precise estimates of a broad distribution into a misleading claim of precise individual prediction.
Probability in a generative account
In SMM/DTT, an inner generation rule may be deterministic, stochastic or described only after a reduction. Each choice should specify where probabilities enter and how they reach completed records. A qualitative appeal to hidden complexity does not determine a probability measure.
The expanded book's reconstruction requirements separates the proposed source from the state, dynamics, probability rule and observable dictionary needed to use it physically. A useful finite test would predict several temporal statistics from the same preparation and parameters, then compare them with an alternative model.
Continue with laws and initial conditions and determinism and predictability to separate the existence of a rule from the information needed to apply it.
Sources and further reading
- Daniel T. Gillespie, Exact stochastic simulation of coupled chemical reactions, Journal of Physical Chemistry 81 (1977), 2340–2361.
- Predictive states: why matching one-step probabilities can leave different future sequences.