[Math Lair] Principle of Indifference

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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 1n.

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 the physical forces that act on a coin when it is flipped are difficult to describe, there is no reason to believe that they favour one side over another, so we can say that the probability of each is ½. The principle of difference 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 0.5. 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 0.5. 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 0.5. 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.

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