1. The Problem. In spite of the vast discussion which has been devoted to the theory of probability, the problem of the short run has received surprisingly little attention. Yet, the whole significance of the theory depends upon a solution of this problem, for without an answer to it we cannot say why it is useful to have knowledge of probabilities or why we should take account of this knowledge in making practical decisions. As far as I know, Charles Peirce was the first to appreciate the importance and difficulty of the question, and his characteristically vivid formulation serves as an excellent starting point for its consideration. He says:

According to what has been said, the idea of probability essentially belongs to a kind of inference which is repeated indefinitely. An individual inference must be either true or false and can show no effect of probability; and therefore, in reference to a single case considered in itself, probability can have no meaning. Yet if a man had to choose between drawing a card from a pack containing twenty-five red cards and a black one, or from a pack containing twenty-five black cards and a red one, and that of a red one were destined to transport him to eternal felicity, and that of a black one to consign him to everlasting woe, it would be folly to deny that he ought to prefer the pack containing the larger proportion of red cards, although from the nature of the risk, it cannot be repeated. [Collected Papers, 2.652]