According to Ludwig von Mises, all human action has to deal with the uncertainty of the future. The distinction between "risk" and "uncertainty" has been developed by Frank Knight and has been more deeply analyzed by Ludwig von Mises, who has shown that they can be subsumed under the more general categories of "class probability" and "case probability". According to Ludwig von Mises and Murray Rothbard, inter alia, only class probability is subject to numerical expression and analysis. This claim has been challenged by Mark Crovelli, who has argued that probability is a subjective measure of uncertainty, and that numerical probability can therefore be legitimately applied to singular cases.
Class probability means, according to Ludwig von Mises, that we know nothing about an individual outcome, but we know everything about a whole class of events, and are certain about the future. In a lottery, for example, we know how many tickets are in total and how many will be drawn. But that does not say at all, if a particular ticket or tickets will win, and buying more tickets does not increase the chance of winning. An instance of class probability is called risk. It is possible to insure against risk, because the behavior of a class of events (or a reasonable subset of it) is well known.
Risk and Insurance
Risk occurs when an event is a member of a class of a large number of homogeneous events and there is fairly certain knowledge of the frequency of occurrence of this class of events. For example, a firm producing bolts knows from long experience that, say, 1 percent of these bolts will be defective. It will not know whether any given bolt will be defective, but it will know the proportion of the total number. This knowledge can be converted into a definite cost of the firm’s operations, especially where enough cases occur within a firm. In other situations, a given loss or hazard may be large and infrequent in relation to a firm’s operations (such as the risk of fire), but over a large number of firms it could be considered as a "measurable" or actuarial risk. The firms could pool their risks, or a specialized firm - an insurance company - could organize the pooling for them.
Case probability means, that we know some of the factors which determine the outcome of a particular event; but there are other determining factors which we don't know. The cases are individual, unique, and nonrepeatable, their result is uncertain. If in roulette a ball falls ten times on red in succession, the probability, that in the next turn will be the result black, is not greater than it was before. Football games cannot be predicted on the results of last games, nor can be presidential elections.
According to Ludwig von Mises, the field for the application of case probability is the field of the sciences of human action, ruled by teleology. It is not subject to numerical expression and analysis. This claim has also been challenged by Mark Crovelli, who has argued that Ludwig von Mises's distinction between "class" and "case" probability is artificial and trivial if one defines probability subjectively 
Probability and Action
Profit and loss are the results of entrepreneurial uncertainty. Risk can be measured or insured against and can be converted into a cost of business operation. It is not responsible for profits or losses - except when the estimates will turn out to be wrong.
Estimates of future costs, demands, etc., on the part of entrepreneurs are all unique cases of uncertainty, where methods of specific understanding and individual judgment of the situation must apply, rather than objectively measurable or insurable risk. Every action is speculation. There is in the course of human events no stability and consequently no safety.
Probability and Forecasting
Men try to forecast particular future event on the basis of their knowledge about the behavior of the class. A doctor may determine the chances for the full recovery of his patient if he knows that 70 per cent of those afflicted with the same disease recover. If he expresses his judgment correctly, he will not say more than that the probability of recovery is 0.7, that is, that out of ten patients not more than three on the average die. This is not a forecast, but a statement about the frequency of the various possible outcomes. All such predictions about external events, i.e., events in the field of the natural sciences, are of this character. They are based either on statistical information or simply on the rough estimate of the frequency derived from nonstatistical experience.
A surgeon tells a patient who considers submitting himself to an operation that thirty out of every hundred undergoing such an operation die. If the patient asks whether this number of deaths is already full, he has misunderstood the doctor's statement. He has fallen prey to the error known as the "gambler's fallacy". Like the roulette player who concludes from a run of ten red in succession that the probability of the next turn being black is now greater than it was before the run, he confuses case probability with class probability.
With case probability, any reference to frequency is inappropriate, according to Ludwig von Mises because the statements always deal with unique events, which are not members of any class. We can form a class of "American presidential elections". It may prove useful or even necessary for various kinds of reasoning. But if we are dealing with the election of, say, 1944, or 2008, we are grappling with an individual, unique, and nonrepeatable case. The case is characterized by its unique merits, it is a class by itself. All the marks which make it permissible to subsume it under any class are irrelevant for the problem in question.
Two football teams, the Blues and the Yellows, will play tomorrow. In the past the Blues have always defeated the Yellows. This knowledge is not knowledge about a class of events. If it was, we would have to conclude that the Blues are always victorious and that the Yellows are always defeated. We would not be uncertain of the outcome of the game, we would know that the Blues will win again. If, on the other hand, we would fall to the "gambler's fallacy," we would argue that tomorrow's game will result in the success of the Yellows.
Gambling and betting
Gambling refers to wagering on events of class probability, such as throws of dice, where there is no knowledge of the unique event. Betting refers to wagering on unique event about which both parties to the bet know something — such as a horse race or a Presidential election. In either case, however, the wagerer is creating a new risk or uncertainty.
According to Rothbard, it is not accurate to apply terms like "gambling" or "betting" to situations either of risk or of uncertainty. They refer to situations where new risks or uncertainties are created for the enjoyment of the uncertainties themselves. Gambling on the throw of the dice and betting on horse races are examples of the deliberate creation of new uncertainties which otherwise would not have existed. The entrepreneur is not creating uncertainties for the fun of it. On the contrary, he tries to reduce them as much as possible. The uncertainties he confronts are already inherent in the market situation, indeed in the nature of human action; someone must deal with them, and he is the most skilled or willing candidate. In the same way, an operator of a gambling establishment or of a race track is not creating new risks; he is neither a gambler or a bettor, he is an entrepreneur trying to judge the situation on the market.
The definition and theory of probability just outlined has been criticized by Mark Crovelli as inconsistent with the Austrian view of causality. On the one hand, Crovelli has criticized Ludwig von Mises for both failing to define probability, and for making ad hoc use of his brother's, Richard von Mises, "frequentist" definition of probability. Crovelli argues that if the world is governed by causal laws, as Ludwig von Mises himself claimed it was, then probability must be defined as a subjective measure of human uncertainty about the causal forces at work in the world. Crovelli claims that if probability is defined as a subjective measure of human uncertainty, then there exists no non-arbitrary reason to restrict its application to "class probability," as Ludwig von Mises claimed.
- Murray N. Rothbard. "9. Risk, Uncertainty, and Insurance", Man, Economy and State online version, referenced 2009-10-24.
- Ludwig von Mises. "2. The Meaning of Probability", Human Action, online version, referenced 2009-10-22.
- Mark R. Crovelli. "On The Possibility of Assigning Probabilities to Singular Cases, or: Probability is Subjective too!", Libertarian Papers Volume 1,#26 (2009), referenced 2011-01-10.
- Ludwig von Mises. "3. Class Probability", Human Action, online version, referenced 2009-10-10.
- Ludwig von Mises. "4. Case Probability", Human Action, online version, referenced 2009-10-10.
- Mark R. Crovelli. "A Challenge to Ludwig von Mises's Theory of Probability", Libertarian Papers Volume 2, #23 (2010), referenced 2011-01-10.
- Probability on Wikipedia
- The limits of numerical probability: Frank H. Knight and Ludwig von Mises and the frequency interpretation (pdf) by Hans-Hermann Hoppe
- Risk, Uncertainty, and Economic Organization by Peter G. Klein
- The Correct Theory of Probability, by Murray N. Rothbard
- What Is the Proper Way to Study Man? by Murray N. Rothbard
- Why the Definition of Probability Matters by Mark R. Crovelli, January 2011