The spelunker cannot predict because of total ignorance of what lies ahead. She is in a situation that, so far as she knows, has never occurred before. She is like a scientist exploring an entirely new avenue of research. The scientist can make educated guesses about what might happen, but can neither say what will happen nor even assess the probability of any particular outcome of the exploration. That is a situation of uncertain knowledge and uncertain probability.
The gambler is in a better position. He has uncertain knowledge but certain probability. He knows all the possible outcomes of his throw and knows exactly the chance that any particular outcome will actually occur. His ignorance of any single result is tempered by a definite knowledge of average results.
The probability of atomic multitudes, which is the same as the probability of the gambler, lies at the heart of some of the most important aspects of the behavior of matter in bulk. This kind of probability we can call a probability of ignorance—not the nearly total ignorance of the spelunker in a new cave or the researcher on a new frontier, but the ignorance of certain details called initial conditions. If the gambler knew with enough precision every mechanical detail of the throw of the dice and the frictional properties of the surface onto which they are thrown (the initial conditions) he could (in principle) calculate exactly the outcome of the throw. Similarly, the physicist with enough precise information about the whereabouts and velocities of a collection of atoms at one time could (with an even bigger “in principle”1) calculate their exact arrangement at a later time. Because these details are lacking, probability necessarily enters the picture.
The prospector’s uncertainty is of still a different kind. He is coming up against what is, so far as we now know, a fundamental probability of nature, a probability not connected with ignorance of specific details, but rather connected with the operation of the laws of nature at the most elementary level. In atomic and nuclear events, such as radioactivity, probability plays a role, even when every possible initial condition is known. This is quantum probability (see Essay Q3). In thermodynamics—the study of the average behavior of large numbers of molecules and of the links between the submicroscopic and macroscopic worlds—the fundamental probability in nature is of only secondary importance. It influences the details of individual atomic and molecular collisions, but these details are unknown in any case. Of primary importance is the probability of ignorance stemming from our necessarily scant knowledge of precise details of molecular motion.
The triumphs of thermodynamics are its definite laws of behavior for systems about which we have incomplete knowledge. However, it should be no surprise that laws of probability applied to large enough numbers can become laws of near certainty. The owners of gambling casinos are consistent winners.
1 Because classical mechanics does not suffice to calculate exactly the outcome of an atomic collision, this hypothetical forecast of future atomic positions and velocities could be extended but a moment forward in time.