Talk:Wine/water paradox

Possible misunderstanding
One of the hazards of a problem like this is that "pure" mathematicians, including many probabilists, will misunderstand it and say, for example, that no well-posed math problem has been identified, which is true but misses the point. Michael Hardy (talk) 20:44, 15 May 2019 (UTC)

dt/t ?
If invariance under the mapping $ t \mapsto 1/t$ is desired (making the roles of water and wine identical), how about the measure dt/t?
 * $$ \int_{1/3}^3 \frac{dt} t = \log 3 - \log \frac 1 3 = 2 \log 3, $$

so
 * $$ \frac{dt}{t\log 3} $$

is a probability measure on Borel subsets of [1/3, 3]. And

\begin{align} & \frac d {du} \Pr\left(\frac 1 x \le u\right) = \frac d {du} \Pr\left( x \ge \frac 1 u \right) = \frac d {du} \int_{1/u}^3 \frac{dt}{t\log 3} \\[8pt] = {} & -\frac 1 {(1/u)\log3} \cdot \frac d {du}\,\frac 1 u = \frac 1 {u\log 3}, \end{align} $$ so x and 1/x are identically distributed. Michael Hardy (talk) 20:56, 15 May 2019 (UTC)


 * Wouldn't it be simpler to let y be uniformly distributed on [¼,¾] and take x = (1 - y)/y, so that 1/x = y/(1 - y)? Then the distribution of 1/x is the mirror image of x (substituting 1 - y for y turns 1 - y into y). Vaughan Pratt (talk) 05:35, 20 March 2021 (UTC)

Wine-alcohol conflation
The parenthetical " (i.e. 25-75% alcohol)" would appear to assume that the wine is 200 proof, i.e. 100% alcohol by volume. Obviously we shouldn't solve this by calling it the alcohol/water paradox. Options I can think of would be to rephrase the parenthetical as "(i.e. 25-75% wine)", or simply delete it altogether. Any preferences? Vaughan Pratt (talk) 04:20, 20 March 2021 (UTC)