Another way of explaining the problem is to think of how the probability of picking the best answer from a set of alternatives varies with the range of alternatives included. In the classic jury theorem case the alternatives are institutionally constrained: innocent or guilty. But suppose the jury then had to decide on a sentence, that the correct answer is 2.5 years in jail, and that the alternatives are (i) 1 year and (ii) 3 years. If jurors are no better than random in this context, they have a bit of a liberal bias, but not much. But now suppose the options are (i) 30 days house arrest (ii) 1 year in jail (iii) 3 years in jail (iv) life in prison. If voters are random with respect to this set of choices then they are grossly incompetent. So the meaning of randomness varies depending on the range of alternatives included (and also the position of the correct answer in that range).

The disjunction problem arises if the alternatives are of different sizes, so to speak. Imagine that the correct answer is a particular point on a line, and that we divide up the line into segments to get our alternatives. The meaning of random competence will vary depending on how much of the line the various segments cover. If we take three equal segments, but then subdivide one of them, we can’t assume random competence both with respect to the original division and the new division. Again, the meaning of random competence depends upon some substantive judgment about the nature of the alternatives.