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Probability
1654· Mathematics
ProvenTested many times; part of working physics.
The mathematics of chance, founded in 1654 letters between Blaise Pascal and Pierre de Fermat about a gambling problem.
See it on the Science map →How it works
Probability assigns each possible outcome a number from 0 (impossible) to 1 (certain) and gives rules for combining them. Pascal and Fermat worked out how to split the pot of an interrupted game fairly. Andrey Kolmogorov gave it firm axioms in 1933. In physics it appears twice: as a tool for handling huge numbers of atoms we can't track individually, and in quantum mechanics as a basic feature of nature.
Why it matters
It turns ignorance and randomness into reliable predictions, from insurance tables to gas pressure to quantum experiments.
Think of it like
An insurer can't predict which building will burn, but across thousands of policies the loss rate is very predictable. Statistical physics treats atoms the same way.
How we know
Mathematically rigorous since Kolmogorov (1933). Its predictions are confirmed in thermodynamics, genetics and quantum experiments, which show outcome frequencies exactly as predicted.
Still a mystery
Whether quantum randomness is truly fundamental, and what probability really means (long-run frequency or degree of belief), are still debated.
Connections (2)
▸Probability lets Boltzmann count states for Entropy: disorder always growsProven
The relationship
Boltzmann couldn't track trillions of molecules individually, so he used probability to ask which overall states are most likely. Entropy became a measure of likelihood: the more ways to arrange the atoms for a given state, the more probable that state. Probability turned thermodynamics' unexplained rule into a near-certain statistical outcome.
How the link was found
Maxwell brought probability into gas theory in 1860; Boltzmann extended it to entropy in 1872–1877; Josiah Willard Gibbs generalised it in 1902.
Why it's proven
Statistical mechanics accurately predicts measured properties of gases, solids and radiation, and small-scale fluctuation experiments match the predicted probabilities.
Open questions and mysteries
Whether these probabilities reflect our ignorance of atomic details or something objective about nature is still debated by philosophers of physics.
▸Probability via the Born rule, extracts predictions from The wave functionProven
The relationship
The wave function itself is never observed directly. The Born rule says the chance of getting a particular result equals the square of the size of the wave function's number for that result. Probability is the bridge from the abstract wave function to what labs actually measure: how often each outcome occurs.
How the link was found
Max Born proposed it in 1926, adding the crucial squaring in a footnote; he received the Nobel Prize for it in 1954.
Why it's proven
Measured outcome frequencies match it in every quantum experiment, including dedicated tests for small deviations, which found none.
Open questions and mysteries
Why the rule takes this form, and whether it can be derived rather than assumed (for example in the many-worlds interpretation), is debated.