Classical Probability

The classical interpretation is the oldest and, for a century and a half, was simply what "probability" meant. Its definition is Laplace's, from the Philosophical Essay on Probabilities (1814):

The probability of an event is the ratio of the number of cases favourable to it, to the number of all cases possible, when nothing leads us to expect that any one of these cases should occur more than any other.

So the probability of rolling an even number is $3/6$: three of six equally possible faces are even. The definition is exactly right for the problems that created the subject — Pascal and Fermat's gambling questions, Huygens' De Ratiociniis in Ludo Aleae, the urn problems of the Bernoullis — and it remains the first thing anyone is taught.

This page sets out the definition and the Principle of Indifference that underwrites it, the circularity charge that has been pressed against it since Boole, the paradoxes that arise when possibilities can be carved in more than one way, and the residue that survives: symmetry is excellent evidence for a probability assignment even though it is not a definition of one.

The definition and its engine

The classical account has two components, and they are worth separating because only the second is contentious.

The first is arithmetic: given a set of cases each carrying equal probability, the probability of an event is the proportion of cases in which it obtains. Nobody disputes this, and it follows immediately from the axioms — it is just the uniform measure on a finite sample space.

The second supplies the equal probabilities in the first place. This is the Principle of Indifference (Keynes's name; Laplace spoke of "equally possible" cases and Bernoulli of "equal facility"):

If there is no reason to prefer any one of a set of mutually exclusive and jointly exhaustive possibilities over another, assign them equal probability.

The principle is epistemic in form — it appeals to the absence of reason to discriminate — which makes the classical interpretation harder to classify than it first appears. Laplace was a determinist who held that probability arises entirely from ignorance, so on his own understanding classical probability is closer to a theory of rational credence than to a theory of objective chance. Read that way it is an ancestor of the logical interpretation, and its modern descendants are on the epistemic side of the chance/credence divide.

The circularity objection

The standard objection is that the definition is viciously circular, and it is very old — Boole pressed it, and von Mises and Keynes each thought it decisive.

"Equally possible" must mean something. If it means equally probable, then probability has been defined in terms of probability, and we have learned nothing. If it means something else — equally possible in some modal sense — then the definition is false, because mere possibility does not come in degrees that could ground a ratio. That a biased coin can land heads and can land tails makes both possible; it does not make them equiprobable.

Two responses are available, and neither is fully satisfying.

• Take equipossibility as primitive, grounded in the physical symmetry of the setup rather than in probability. This is the more promising line and is developed below, but it concedes that the definition is not self-standing: it now depends on a prior notion of symmetry that must itself be established.
• Take the principle as a rationality constraint rather than a definition — a rule about what to believe under ignorance, not an account of what probability is. This is the honest retreat, and it converts the classical interpretation into a normative thesis about priors, which is where the objective Bayesian inherits it.

Either way, the classical account is not an interpretation of probability in the sense the other positions offer. It tells us how to compute probabilities in a restricted class of cases once a symmetry is granted; it does not tell us what a probability is.

The paradoxes of indifference

The deeper problem is not circularity but inconsistency. The Principle of Indifference gives different answers depending on how the possibilities are described, and since it offers no way to privilege a description, it can be made to yield contradictions.

The wine–water paradox. A mixture of wine and water has a ratio of wine to water somewhere between $1/3$ and $3$. Applying indifference to that ratio, the probability that it is at most $2$ is $(2-1/3)/(3-1/3)=5/8$. But now consider the ratio of water to wine, which lies between $1/3$ and $3$ as well. The event "wine/water $\le 2$" is the event "water/wine $\ge 1/2$", whose probability by indifference on the second ratio is $(3-1/2)/(3-1/3)=15/16$. Two applications of the same principle to the same physical situation, differing only in which quantity is called the variable, give $5/8$ and $15/16$.

The cube factory. A factory makes cubes with side length between $0$ and $1$ metre. What is the probability that a cube has side at most $1/2$? By indifference over side length, $1/2$. But the same cube has face area between $0$ and $1$ square metre and volume between $0$ and $1$ cubic metre; indifference over area gives $1/4$, and over volume $1/8$. The three descriptions are inter-definable and equally natural, and the principle cannot choose.

Bertrand's paradox is the continuous case in its most famous form, giving $1/3$, $1/2$, or $1/4$ for the same chord problem depending on the randomisation.

The common structure is that indifference is not invariant under reparameterisation. A uniform distribution over a quantity is not uniform over a non-linear function of that quantity, and nothing in the principle says which parameterisation is the right one. Since the principle's whole appeal was that it required no information, and the choice of parameterisation is precisely information, the appeal is illusory.

There is also a discrete version, less often noted but equally damaging. Confronted with a coin of unknown bias, indifference over $\{$heads, tails$\}$ gives $1/2$ each. But indifference over the hypotheses about the bias — that the coin is fair, two-headed, or two-tailed — gives $1/3$ each, from which the probability of heads is $1/2$ again only by luck of the example; other partitions of hypothesis space give other answers. Which partition is the one the agent is ignorant across?

What survives: symmetry as evidence

The paradoxes refute indifference as a general rule. They do not touch the observation that got the subject started: for a fair die, $1/6$ really is the right answer, and it is not a coincidence that the classical method works.

What distinguishes the successful cases is that the symmetry is physical rather than merely descriptive. A well-made die is invariant under a group of rotations that permutes its faces; that invariance is a fact about the object, testable by inspection and by rolling. The cube factory has no analogous invariance — nothing about the manufacturing process privileges length over volume — and the wine–water mixture has none either. Where a genuine symmetry of the chance setup maps outcomes onto one another, the assignment of equal probabilities is grounded in the world; where the "symmetry" is only a feature of how we happened to describe the situation, it is grounded in nothing.

This distinction can be pushed further, and it is one of the more satisfying results in the area. Poincaré's method of arbitrary functions, developed by Hopf and later by Engel and von Plato, shows why symmetric mechanical devices produce stable frequencies. Consider a roulette wheel spun with initial angular momentum drawn from some density $f$. If the wheel's dynamics maps momentum to final position with rapid oscillation, then for any reasonably smooth $f$ — the croupier's habits do not matter — the induced distribution over red and black converges to $1/2$ each. The uniformity is a consequence of the dynamics plus a mild smoothness assumption, not of an assumption of ignorance.

This is a genuine vindication of the classical intuition, and it changes its status. The equal probabilities are derived from physical facts about the device rather than postulated from a lack of information, which is exactly what the circularity objection demanded. It also explains the pattern of successes and failures: the classical method works for dice, coins, roulette wheels, and well-shuffled cards — all devices deliberately engineered to have the relevant symmetry — and fails everywhere else.

Assessment

Against the criteria of adequacy:

The verdict is that the classical interpretation is not a general theory of probability but a special case with an unusually good justification. Where a physical symmetry group acts on the outcome space, equal probabilities are the right assignment and the method of arbitrary functions explains why. Where there is no such group, the principle that generated the answers generates contradictions instead.

Where this sits

Two threads run forward from here, and both are live.

The indifference principle is not dead but demoted. Stripped of its role as a definition, it survives as a proposal about how to choose priors under ignorance — a constraint on rational credence rather than an account of what probability is. Jaynes's transformation-group and maximum-entropy methods are the sophisticated form of that proposal, and they attempt exactly the repair this page has sketched: replace "no reason to discriminate" with "invariance under a specified group". Whether that succeeds is the question of objective Bayesianism, and the general difficulty of choosing a parameterisation is taken up again under priors and indifference.

The circularity problem — that the account presupposes a notion it should explain — recurs in a different key in every interpretation that follows. Frequentism must say which sequence, propensity must say what a tendency is, and subjectivism must say what a degree of belief is. The classical account is the first and clearest instance of a pattern: interpretations of probability tend to succeed by relocating the primitive rather than eliminating it.

The next page takes up the most ambitious attempt to make probability fully objective on the epistemic side: the logical interpretation, which treats probability as a relation of partial entailment and inductive inference as a branch of logic.