the loss of the letter) which has certainly happened; and we suppose that, of the only two causes to which it can be assigned, the ‘value,’ i.e.
statistical frequency, of one is accurately assigned, does it not seem natural to suppose that something can be inferred as to the likelihood that the other cause had been operative? To say that nothing can be known about its adequacy under these circumstances looks at first sight like asserting that an equation in which there is only one unknown term is theoretically insoluble.
As examples of this kind have been amply discussed in the chapter upon Inverse rules of Probability I need do no more here than remind the reader that no conclusion whatever can be drawn as to the likelihood that the fault lay with my friend rather than with the Post Office. Unless we either know, or make some assumption about, the frequency with which he neglects to answer the letters he receives, the problem remains insoluble.
The reason why the apparent analogy, indicated above, to an equation with only one unknown quantity, fails to hold good, is that for the purposes of Probability there are really two unknown quantities. What we deal with are proportional or statistical propositions. Now we are only told that in the instance in question the letter was lost, not that they were found to be lost in such and such a proportion of cases. Had this latter information been given to us we should really have had but one unknown quantity to determine, viz.
the relative frequency with which my correspondent neglects to answer his letters, and we could then have determined this with the greatest ease.
[1] Discussed by Mr F. Y. Edgeworth, in the Phil.
Mag.
for April, 1887.
[2] Journal of the Statistical Soc.