Principle of Indifference

The principle of indifference, also known as the principle of insufficient reason, is a common principle applied to events in calculating their probabilities. It was first described by James Bernoulli. One way of putting it is as follows: If there are n possible outcomes in a situation, and there is no reason to believe that any possibility is more or less likely than any other, then the probability of each is ¹⁄_n.

Standard examples of the principle of indifference include flipping a coin, rolling a die, or drawing a card from a deck. For example, a coin can land on heads or tails. While it's difficult to describe the physical forces acting on a coin when it is flipped, there is no reason to believe that they favour one side over another, so we can say that the probability of each outcome is ½. The principle of indifference is also useful in the field of subjective probability.

The principle of indifference needs to be treated with caution. Say that we are given a square, and we are told that the length of its side is between 3 inches and 5 inches. We don't have any reason to believe that any side length is more likely than any other. So, by the principle of indifference we might say that the probability of the length of its side being between 3 inches and 4 inches is ½. However, we can also look at the area of the square. The area of the square is between 9 square inches and 25 square inches. Since the area could be any number in between there, and we have no reason for believing any number to be more likely, we could state that the probability of the area being between 9 and 17 square inches is ½. However, that doesn't make sense because we just said that the probability of its side length being no more than 4 inches, and hence its area no more than 16 square inches, is ½. In this case, assigning a probability to the length of the side is meaningless unless we have more information on how the square is constructed.

Here's a similar example. A certain family has two children. You are told that at least one is a boy. What is the probability that both are boys? One way of looking at the problem is that there are two possibilities: Either the family has one boy and one girl, or it has two boys. Each possibility is equally likely, so the probability that both are boys is ½. Another way of looking at the problem is that there are three possibilities: The older is a boy and the younger is a boy, the older is a boy and the younger is a girl, or the older is a girl and the younger is a boy. Each of the three possibilities are equally likely, so the probability that both are boys is ⅓. This is the correct way of looking at the problem. We need to know how families are constructed in order to correctly apply the principle of indifference.

Finally, keep in mind that using the principle of indifference to determine probabilities in scenarios where there is some reason to think that not all possibilities are equally likely will probably result in incorrect results. For pages related to this point, see Monty Hall Problem and Doomsday Argument.

Sources used (see bibliography page for titles corresponding to numbers): 44.