The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or "degrees of belief") equally among all the possible outcomes under consideration. It can be viewed as an application of the principle of parsimony and as a special case of the principle of maximum entropy. In Bayesian probability, this is the simplest non-informative prior.
Examples The textbook examples for the application of the principle of indifference are coins, dice, and cards.
Coins A symmetric coin has two sides, arbitrarily labeled heads (many coins have the head of a person portrayed on one side) and tails. Assuming that the coin must land on one side or the other, the outcomes of a coin toss are mutually exclusive, exhaustive, and interchangeable. According to the principle of indifference, we assign each of the possible outcomes a probability of 1/2. It is implicit in this analysis that the forces acting on the coin are not known with any precision. If the momentum imparted to the coin as it is launched were known with sufficient accuracy, the flight of the coin could be predicted according to the laws of mechanics. Thus the uncertainty in the outcome of a coin toss is derived (for the most part) from the uncertainty with respect to initial conditions. This point is discussed at greater length in the article on coin flipping.
Dice A symmetric die has n faces, arbitrarily labeled from 1 to n. An ordinary cubical die has n = 6 faces, although a symmetric die with different numbers of faces can be constructed; see Dice. We assume that the die will land with one face or another upward, and there are no other possible outcomes. Applying the principle of indifference, we assign each of the possible outcomes a probability of 1/n. As with coins, it is assumed that the initial conditions of throwing the dice are not known with enough precision to predict the outcome according to the laws of mechanics. Dice are typically thrown so as to bounce on a table or other surface(s). This interaction makes prediction of the outcome much more difficult. The assumption of symmetry is crucial here. Suppose that we are asked to bet for or against the outcome "6". We might reason that there are two relevant outcomes here "6" or "not 6", and that these are mutually exclusive and exhaustive. A common fallacy is assigning the probability 1/2 to each of the two outcomes, when "not 6" is five times more likely than "6."
Cards A standard deck contains 52 cards, each given a unique label in an arbitrary fashion, i.e. arbitrarily ordered. We draw a card from the deck; applying the principle of indifference, we assign each of the possible outcomes a probability of 1/52.
Criticism The deck-of-cards example, more than the others, shows the difficulty of actually applying the principle of indifference in real situations. What we really mean by the phrase "arbitrarily ordered" is simply that we don't have any information that would lead us to favor a particular card. In actual practice, this is rarely the case: a new deck of cards is certainly not in arbitrary order, and neither is a deck immediately after a hand of cards. In practice, we therefore shuffle the cards; this does not destroy the information we have, but instead (hopefully) renders our information practically unusable, although it is still usable in principle. In fact, some expert blackjack players can track aces through the deck; for them, the condition for applying the principle of indifference is not satisfied. In a macroscopic system, at least, it must be assumed that the physical laws that govern the system are not known well enough to predict the outcome. As observed some centuries ago by John Arbuthnot (in the preface of Of the Laws of Chance, 1692),
It is impossible for a Die, with such determin'd force and direction, not to fall on such determin'd side, only I don't know the force and direction which makes it fall on such determin'd side, and therefore I call it Chance, which is nothing but the want of art.... Under the principle of causal determinism, given enough time and resources, there is no fundamental reason to suppose that suitably precise measurements could not be made, which would enable the prediction of the outcome of coins, dice, and cards with high accuracy: Persi Diaconis's work with coin-flipping machines is a practical example of this.
Application to continuous variables Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables. A typical case of misuse is the following example:
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