The probability of rolling a 'five' with an ordinary die is 1/6. Therefore, it follows that the proposition 'the 3rd roll of an ordinary die will be a "five"' has a 16% probability of being true, right? Well, not exactly. It may seem quite a reasonable conclusion, but there a problem lurking away from plain sight. The purpose of this thread is to bring that problem into the light.

If the probability of rolling a 'five' with an ordinary die is 1/6 then it would be sensible to predict 'five' to turn up once for every six rolls of the die. However, if the proposition 'the 3rd roll of an ordinary die will be a "five"' has a 16% probability of being true, is it sensible to predict that the same 6th roll of the die will turn up a 'five' once for every six same 6th rolls of the die? That is a contradiction. The same six throws would not be the same if only one turned up a six, and the only way to resolve the contradiction is if all of the six same throws turn up a 'five' or some other number. Therefore, the proposition 'the 3rd roll of an ordinary die will be a "five"' can only have a probability of either 1 or 0. In other words, propositions are never probably true.

For example, imagine that someone was tasked with desgining a fleet of aircraft. That person surveyed the scientific literature and discovered the theory of gravity. However, he also noted a consensus between scientists that this theory is only probably true, about 80% probable. Thinking on this the aircraft designer came to a decision: he would design his aircraft so that they would not only function correctly according the theory of gravity, but also function correctly according to an alternative, less probable, theory of gravity. Afterall, would it not be sensible to predict that the prevailing theory of gravity will be false for two of every ten flights?

That said, after more scientific evidencemakesthe theory more probable, perhaps 90%, it would be sensible to only expect the theory to be false for one of every ten flights, right?

The strange thing is that improbable theories are actually the best. The most probable theories are tautological, and utterly useless. For example, consider again the ordinary die that is to be rolled, and the following two predictions.

P1: the next roll will be a one.*

P2: the next roll will be a one, two, or three.

Now which of these predictions is the most probable? Well, the probability of P1 is 1/6, whereas the probability of P2 is 1/2. In short, P2 is more probable, but it is also the least informative. P1, if true, is much preferrable to P2, since it is far more specific. Moreover, note that improbable theories are not only more informative, but usually more falsifiable i.e. experimentally testable.

Okay, that's enough, my incomprehensible rant is over...

* The word 'next' in the statement 'the next roll will be a one' is a varible, which makes the statement an unfinished proposition i.e. it does not say anything about the world until the 'next' is specified. It is interesting to note that until the variable is satisfied we can sensibly talk about it with probabilities, since we are not actually asserting anything about the world