There are two kinds of probability: a priori and a posteriori. The first will be discussed here, while the second will be discussed in a separate thread. Without relying on experience, the truth of a priori probability can be known by logical arguments using mutually agreed definitions, for example, tossing an ideally “fair” number cube whose probabilities of showing any number from 1 to 6 are deduced from the principle of equally likely outcomes. This works if and only if the cube is truly “fair” in every sense. However, for all practical purposes, the construction of a cube with the number 1 opposites the number 6, 2 opposites 5, and 3 opposites 4 such that sums of opposites always add up to 7, can be logically argued in favor of equally likely outcomes. On the one hand, it can also be argued that the six markings on all six sides cannot all be symmetrical. A 6 is always more complex than a 1. But a 5 is just the mirror image of a 2. A 3 can be marked as heavy as a 4. However, if the 1 is marked as heavy as a 6 then the cube become potentially a true “fair” solid.

On the other hand, if all six numbers (1 to 6) are arranged at the 3 vertices and 3 midpoints of an equilateral triangle such that 4, 5, and 6 are found at the vertices, while 1, 2, and 3 are found at the midpoints then it can be arranged such that the sum of each side add up to 12. However, if 1, 2, and 3 are found at the vertices, while 4, 5, and 6 are found at the midpoints then the sum of each side add up to 9. These two arrangements cannot represent equally likely outcomes to give two different equal sums of 9 and 12.