Wine/water paradox

For the similarly-named fluid mixing puzzle, see Wine/water mixing problem.

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 $x$ lying in the interval $1/3\leq x\leq 3$. We seek the probability, $P^{\ast }$ say, that $x\leq 2$.

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

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

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

Now consider $y=1/x$ the ratio of water to wine. Again using the Principle of indifference, we get

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

Taking $a=1/2$ we conclude that

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

But since $y=1/x$, we get

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

a paradox.