What is Probability?

QM adds confusion to what one may worry is already a complicated issue in classical probability theory, namely how we should think about probabilities. The inventors of the theory, particularly those whose primary interest was in gambling or other financial transactions, clearly thought of it as a sophisticated way of guessing an unpredictable future. This interpretation is clearly tied up with human psychology. A discussion of probability from this point of view, which attempted to makevery precise rules about how to guess and how to use additional data to assess and improve the quality of one’s guesses, was given by Bayes in the 18th century. In modern times, this view of probability is given the label Bayesian interpretation.
If you are a gambler or a financier, this is certainly the way you think of probability. Experimental physicists and theorists who follow their work closely also use Bayesian reasoning quite frequently. Looking at experimental lectures you will often see plots, which include lines indicating the predictions of a theory, and colored stripes following the lines, which indicate things like “the 95% confidence interval.” Translated into full English sentences, this means the region of the graph where, with 95% probability the data actually lie, given all the possible random and systematic errors. In graphs referring to the behavior of microscopic systems, these errors include the fact that the quantum theory does not make definite predictions for the number of events of a certain type, but only predicts (see below) the ratio of the number of events to the number of runs of the experiment in the limit that the number of runs goes to infinity. There may also be different bands telling you that the theoretical calculation has “uncertainty,” but this is a completely different sort of error and stems from the fact that we can usually solve the equations of QM only approximately. The use of the word confidence interval in this context is the reflection of the Bayesian outlook on the meaning of probability.
As probability theory became more and more important in science, scientists searched for a more “objective” way of thinking about it, which removed the human psyche and words like confidence from the unbiased description of nature by the combination of mathematics and observation. This led to what is called the frequentist interpretation of probability. According to this paradigm, you test a prediction which gives probabilistic answers by repeating your experiment N times, and recording the fraction of times the experiment produces each possible result. As N → ∞ , these fractions converge to the predicted probabilities if the theory in question is correct. This is indeed an objective definition of probabilities, but it is problematic, because it is impossible to take N to infinity, even if the universe lasts for an infinite amount of time. To illustrate the problem, flip a coin 2000 times and observe that it always comes up heads. The probability for that, assuming the coin is unbiased is 2⁻²⁰⁰⁰ ∼ 10⁻⁵⁰⁰, a pretty small number, but this does not prove that someone has weighted the coin. If you think it does, would you bet your life on it?Would you bet the lives of all your loved ones? Would you bet the lives of the entire human race? Obviously, these questions all have subjective answers, which depend on who you are and what your mood is. This is the reason that experimental physicists, who test probabilistic predictions (or even definite predictions that are tested with imprecise machinery) by applying the frequentist rule with a finite number of trials, cite their results in terms of a Bayesian confidence interval. We can try very hard to be completely objective about the data, but no finite amount of effort can completely eliminate the need for “leaps of faith.”

These interpretational problems have nothing to do with QM. They would be there for a completely deterministic theory about which we were ignorant of some of the initial data, and since some of the initial data have to do with the performance of the measuring apparatus itself, or external influences interfering with the machinery (cosmic rays, sound waves, the electromagnetic field generated by a radio 4 km away, etc.), we are always ignorant of some of the data. We continue to do experimental and theoretical physics despite these obstacles, because we believe that we can control these sources of error well enough that we are happy with the small size of the required leap of faith.
The interpretation of QM as a new kind of probability theory is certainly correct, and is the only interpretation that has been tested by experiment. If it is the final word on how to interpret the mathematics, then we will just have to live with the intrinsically indefinite nature of probabilistic predictions. We will explore alternative explanations in Appendix A.