The Australian Mathematical Society

The so-called Wine/Water Paradox is one of a number of puzzles associated with elementary probability theory. It has recently been the subject of renewed interest with the proposal of two competing attempts at resolution. The origins of the paradox will be explored, as will some of its consequences for the understanding of the nature of probability. It will then be argued that the attempted resolutions do not carry the force their proposers attach to them. 1 Statement of the Paradox The Wine/Water Paradox has been given in a number of forms, differing mostly in thenumerical values of the various parameters involved. For convenience, I adopt those givenin a recent re-examination of it [10]. A mixture is known to contain a mix of wine and water in proportions such thatthe amount of wine divided by the amount of water (both measured in terms ofsome agreed unit) is a ratio x lying in the interval 1/3 ≤ x ≤ 3. We seek theprobability, P ∗ say, that x ≤ 2. The problem is, in one sense, insoluble, because insufficient information is supplied toenable a rigorous solution. However, some would argue, according to the so-called Princi-ple of Indifference, that, failing such information, we should suppose that x is uniformlydistributed. That is to say that for any number a in the given interval,