There is here, surely, a rather awkward transition from the “objective” to the “subjective” point of view. We were dealing, in the first case, with groups or series of events about which the doctrine of chances enabled us to say something positive, something which experience would always confirm if the groups or series were large enough. A perfect calculator, endowed with complete knowledge of all the separate group members, would have no correction to make in our conclusions. His information would be more complete than our own, but not more accurate. It is true that for him “averages” would have no interest and “chance” no meaning. Nevertheless, he would agree that in a long series of fair throws of a fair die any selected face would turn up one-sixth times as often as all the others taken together. But in the second case this is no longer so. Foresight based on complete knowledge would apparently differ from foresight based on the calculation of chances. Our calculator would be aware of the exact manner in which the die was loaded, and of the exact advantage which this gave to certain numbers. He would, therefore, know that in asserting the chance of any particular number turning up on the first throw to be one-sixth, we were wrong. In what sense, then, do we deem ourselves to have been right? The answer, I suppose, is that we were right _not_ about a group of throws made with _this_ loaded die, but about a group of such groups made with dice loaded at random—a group in which “randomness” was so happily preserved among its constituent groups that its absence within each of these groups was immaterial, and no one of the six alternative numbers was favoured above another. A similar reply might be given if we suppose our ignorance carried yet a step further. Instead of knowing that our die was loaded, and being ignorant only of the manner of its loading, we might be entirely ignorant whether it was loaded or not. The chances of a particular number turning up on the first throw would still be one-sixth. But the series to which this estimate would refer would neither be one composed of fair throws with a fair die, nor one composed of a series of throws with dice loaded at random, but one composed of a series of throws with dice chosen at random from a random collection of dice, loaded and not loaded!