Paradoxes of Probability
Problem
A popular (but, for reasons to emerge shortly, controversial) principle about probability is:
The Principle of Indifference (PI): Given two alternative possibilities, if there is no reason for preferring either one over the other, then the two are equally probable.
The notion of probability in play here is epistemic (roughly, P(A) is a measure of how much reason we have to believe A).
Example: You know that Sue took a trip of 100 miles, and it took between 1 and 2 hours. Equivalently, you know that the distance was 100 mi, and the average velocity was between 50 and 100 mph. This is all you know. Now apply the PI to the following:
Q1: What is the probability that the trip lasted between 1 and 1.5 hours?
Answer: (1.5 – 1)/(2 – 1) = ½.
Q2: What is the probability that the velocity was between 66.7 and 100 mph?
Answer: (100 – 66.7)/(100 – 50) = 33.3/50 = 2/3.
Problem: Q1 and Q2 are actually the same question. If the trip lasted 1 hour, then the velocity was (100/1) = 100 mph; if it lasted 1.5 hours, then the velocity was (100/1.5) = 66.7 mph. So we know that 1<t<1.5 iff 66.7<v<100. The one cannot have probability ½ and the other have probability 2/3.
There are many other examples like this. You take different ways of describing the possibilities, try to apply the PI, and get incompatible results.
A Bad Response
Some say this means the PI is inconsistent, and thus it should just be completely rejected. What is wrong with that?
The PI is highly intuitive. Suppose you are offered a choice between two identical-looking envelopes. One has $100 in it, and the other has $1 in it. That's all you know. If you pick one of them, what is the probability that you get $100? Everyone knows the answer to this: it is 1/2.
The PI seems analytic, given an epistemic interpretation of probability: if there is no reason to favor A over B or vice versa, then we have exactly as much reason to believe A as B.
The PI lies behind the most reliable, accepted methods of assessing probabilities, those based on statistical evidence and randomization procedures.
Example: After shuffling a deck cards, there's a 1/52 probability that, say, the Jack of Hearts is on top. This is about the least controversial probability assessment you can make. This probability is an application of the PI. There are many different possible deck-shufflings that would be indistinguishable to you (including different starting positions for the deck and slightly different ways you could have moved the cards). Some of these would put the jack of hearts on top. Some would put the 4 of clubs on top. Etc. You don't have any more reason to think that a jack-of-hearts-shuffling occurred rather than a 4-of-clubs-shuffling, etc. So you assign an equal probability to each type of shuffling. A similar point applies to all standard randomization procedures.
Similarly for all uses of statistical evidence to assess probabilities. They're all applications of the PI. Say you take a survey of 1000 random Americans and find that 52% of them believe abortion is wrong. You conclude that probably about 52% of Americans oppose abortion (give or take a few %). The basis for this is the fact that, of all the ways of taking a sample of 1000 people out of the U.S. population, the great majority of ways would result in a sample whose proportion of anti-abortionists was close to the proportion in the general population. Roughly speaking, you treat each way of sampling as equally probable, whence you conclude that this sample probably roughly matches the general population. Without the PI, you can't say this.
So if you don't accept the PI you should probably reject all probability assessments (including this one).
After doing that, you should reject all scientific knowledge, and all knowledge of everything not immediately observed by you, because all of that is only probabilistic knowledge.
Solution to the Time/Velocity Puzzle
The correct application of the PI must take account of explanatory priority. The velocity at which you traveled explains how much time the journey took (given a fixed distance); the duration of the trip doesn't explain your speed.
Therefore, the PI should be applied to the range of velocities, not the range of times. Thus, the correct probability is the 2/3.
For more, see Paradox Lost, ch. 8.