Wine-Water Paradox

EDIT: The website auto-formats the number lines and I unfortunately don’t know how to get rid of this. You’ll just have to imagine the numbers spread out on each node of the line.
———–
In philosophy, the Principle of Insufficient Reason (PIR) claims that if a person does not have any evidence that would lead them to believe one outcome is more likely than another, they should assign those outcomes equal probabilities. A simpler way of saying this is that in the absence of all evidence, one should assign all possible outcomes equal probability.

For instance, say I tell you that I’m going to wear either a blue, red, or brown shirt today, and want to make a bet with you in which you try to guess which shirt color it will be (assume I’ve already picked out the shirt, and can’t change it after you’ve guessed). What odds would you give to each of the three shirts? The PIR would tell you to give each 33(.333…)% odds of being chosen.

This can also be applied to continuous numbers. Say I’ve written down a number between 0 and 1. What are the odds my number is above 0.5? On a number line:

0 0.5 1
|_______________|_______________|

You can see that half the area is to the left of 0.5, and half is to the right, so the PIR would say the odds are 50%.

This strikes a lot of people as intuitive, and they are inclined to agree with it a priori, but the PIR is not popular among philosophers today because of several paradoxes that seem to result from it. Maybe the most famous (among philosophers who think about this stuff) is the ‘Wine-Water Paradox’:

I’m filling a pitcher with some combination of wine and water, and we know that there will be no more than 3 times of one ingredient than the other (so you can’t have 3.1 times as much wine as water, and visa versa). What are the odds of having more than 2 times as much wine as water (so the ratio of wine to water would be between 2/1 and 3/1)?

If you look at the number line:

0 1 2 3
|_______________|_______________|_______________|
|
1/3

So the odds of the ratio being between 2/1 and 3/1 are 1/(2+2/3), or approximately 37.5%.

But notice that asking if the ratio of wine to water is between 2/1 and 3/1 is equivalent to asking if the ratio of water to wine is between 1/2 and 1/3.

One the number line now:

0 1 2 3
|_______________|_______________|_______________|
| |
1/3 1/2

the odds become (1/6)/(2+2/3) or approximately 6.25%.

So asking the same question gives different answers according to the PIR, which cannot be.

The solution to this paradox lies in not using an ‘absolute’ number line like the ones above, but what might be called a ‘ratio’ number line:

3:1 2:1 1:1 1:2 1:3
|__________|__________|__________|__________|

By using this ratio line, the answer to our question becomes 1/4, either way you ask the question, thus solving the paradox.

Still this shouldn’t be construed as proof of the PIR, as there are more paradoxes out there, and even the few philosophers who do argue for it do not do so in all circumstances (e.g. you cannot use it on yes/no questions like ‘Is there a God?’).

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