The wine-water paradox is a hoax. Note that the wine-water paradox is supposedly a way of demonstrating that the principle of insufficient reason does not contain sufficient information to choose a prior decision in Bayesian hypothesis testing. The canonical presentation relies on the ratio of wine to water in a mixed solution, but it makes a fatal error in choosing a uniform probability distribution over the interval of the quotient, rather than the logarithm of the quotient. Taking the logarithms of the ratio resolves the paradox.
- r = wine / water
- s = water / wine = 1 / r
- 1/3 <= r, s <= 3
- P(r <=2) = (2-1/3) / (3-1/3) = 5/8
- P(s >=1/2) = (3-1/2) / (3-1/3) = 15/16
Resolution using logarithms:
- P(r <= 2) = (log 2 – log 1/3) / (log 3 – log 1/3)
- P(s >= 1/2) = (log 3 – log 1/2) / (log 3 – log 1/3)
- These are equal. Paradox resolved.
This apparent paradox is just a numerical artifact of an erroneous assumption about the uniform probability distribution of a quotient.