Wine/water paradox

The wine/water paradox is an apparent paradox in probability theory. It is stated by Michael Deakin as follows:

A mixture is known to contain a mix of wine and water in proportions such that the amount of wine divided by the amount of water is a ratio ${\displaystyle x}$ lying in the interval ${\displaystyle 1/3\leq x\leq 3}$. (i.e. 25-75% alcohol) We seek the probability, ${\displaystyle P^{\ast }}$ say, that ${\displaystyle x\leq 2}$. (i.e. less than or equal to 66%.)

The core of the paradox is in finding consistent and justifiable simultaneous prior distributions for ${\displaystyle x}$ and ${\displaystyle 1/x}$.[1] More precisely, the paradox is derived as follows. We do not know ${\displaystyle x}$ and therefore, using the principle of indifference, we assume that ${\displaystyle x}$ is uniformly distributed, i.e. that

Prob${\displaystyle \{x\leq a\}={\frac {1}{8}}(3a-1)}$.

Taking ${\displaystyle a=2}$ we conclude that ${\displaystyle P^{*}=5/8}$

Now consider ${\displaystyle y=1/x}$ the ratio of water to wine. Again using the principle of indifference, we get

Prob${\displaystyle \{y\geq a\}={\frac {3}{8}}(3-a)}$.

Taking ${\displaystyle a=1/2}$ we conclude that

Prob${\displaystyle \left\{y\geq {\frac {1}{2}}\right\}={\frac {15}{16}}}$.

But since ${\displaystyle y=1/x}$, we get

${\displaystyle {\frac {5}{8}}=}$ Prob${\displaystyle \{x\leq 2\}=P^{*}=}$ Prob${\displaystyle \left\{y\geq {\frac {1}{2}}\right\}={\frac {15}{16}}}$,

a paradox.

References

1. Deakin, Michael A. B. (December 2005). "The Wine/Water Paradox: background, provenance and proposed resolutions". Australian Mathematical Society Gazette. 33 (3): 200–205.