Tim Ludwig, Philosophy for Physics, 31 December 2025.
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Abstract
Even though probability is a key concept in modern physics, its precise meaning is surprisingly unclear. While experts can often rely on their intuitive understanding, intuition is of little use to those who want to learn probability theory. To resolve some confusion around the concept of probability, in this article, we separate the mathematical concept of probability from its scientific applications. As a result, we can circumvent much of the debate around its meaning and give a clear definition of probability. Having a clear definition, in the long run, will allow us to gain a better understanding of the applications of probability theory.
Introduction
Probability is a key concept at the core of statistical physics and quantum mechanics. Despite its importance, or maybe because of it, there is still quite some debate around the meaning of probability. Is it subjective or objective [1-3]? Is it determined a priori or a posteriori [1-4]? Should it be understood as a limit of relative frequencies, as a propensity towards some events in individual runs of a random experiment, or as a degree of believe in some proposition [2,3]? Unfortunately, the debate around these questions seems to be “acrimonious and unproductive” and maybe even “misguided,” because different types of problems might be confused [2]. One way around the debate is to resort to intuition and simply insist that “there is no way of making probability theory correspond to reality without requiring a certain degree of intuition” [5]. While relying on intuition can be a viable approach for experts, it is of little use to those who want to learn about the theory of probability and its scientific applications. Here, rather than resorting to intuition, we discuss another way to circumvent the unproductive debate around the meaning of probability.
The debate and confusion around the concept of probability, I believe, originates from a lack of separation between the mathematical definition on the one hand and its scientific applications on the other hand. Consequently, in this article, we will work towards a definition of probability as a purely mathematical concept. While we will use random experiments, random events, and relative frequencies as a motivation towards the axioms of a theory of probability, we will be careful to separate between the motivation for and the definition of the concept of probability. This careful separation occasionally requires some cumbersome phrasing, which might be a reason why it is usually avoided [6]. Nevertheless, separating probability from its interpretation is well worth the effort, as it allows us to sidestep the debate and give a clear meaning to the concept of probability.
After introducing some basic concepts around random events and relative frequencies in section 1, we motivate the axioms of an elementary theory of probability from basic properties of relative frequencies in section 2.
1. Random events and relative frequency
The description or prediction of relative frequencies of random events is one of the most important applications of the concept of probability. So, to motivate the axioms of probability theory, we will start by discussing random events and their relative frequencies as they occur in random experiments.
1.1. Random events, random experiments, and outcomes
An event is something that happens or occurs under some complex of conditions [7]. For example, a book hits the ground after being dropped, a coin shows heads after being tossed, an atom of a radioactive element decays within a specified time frame after being isolated.
Assuming the conditions are specified, we speak of a certain event, if it occurs certainly under those conditions; and we speak of a random event, if it occurs sometimes and sometimes not under those conditions [7]. Random events can be further distinguished into subclasses: we speak of relative randomness, if an event is random relative to our knowledge; we speak of absolute randomness, if an event is and remains random no matter how much additional knowledge we gain [8]. The examples from above would typically be classified as follows. The ‘book hits the ground’ is a certain event, because we know in advance that it will hit the ground when dropped [9]. The ‘coin shows heads’ is a relative-random event, because we do not know in advance which side will be shown after the coin toss, but it would be possible to know the outcome in advance, if we had just enough knowledge about the initial conditions of the coin toss. The ‘atom decaying in a specified time frame’ is an absolute-random event, because there is—to our knowledge so far—no way to know whether or not such an atom will decay within any specific time frame. The distinction between relative randomness and absolute randomness is relevant to understand how quantum mechanics differs from classical mechanics. For the remainder of this article, however, we will not need this distinction and consider random events in general.
A random experiment is a realization of a complex of conditions under which random events occur [10]. At least in principle, it should be possible to repeat a random experiment arbitrarily often [11]. A single performance or execution of a random experiment is called a trial, a test, or a run of that experiment [10, 12]. For example, the act of tossing a coin is a random experiment, as it can be repeated arbitrarily often and creates a complex of conditions under which random events occur. An individual coin toss, then, is a run of this random experiment.
The possible results belonging to a random experiment are called outcomes [12, 13]. In one run of a random experiment, we observe one and only one outcome as a result [14]. For example, when tossing a coin, the outcomes are \(\mathrm{heads}\) and \(\mathrm{tails}\); and, when rolling a die, the outcomes are \(1, 2, 3, 4, 5, 6\). The random events corresponding to the outcomes are called elementary events. As an immediate consequence of this definition, elementary events are mutually exclusive for one run of a random experiment. For example, when rolling a die, ‘the die shows \(2\)’ is an elementary event, while ‘the die shows \(4\)’ is a different elementary event; and, in a single run, these elementary events cannot occur together. In general, however, multiple events can occur together in one run [14]. For example, when rolling a die, ‘the die shows a number above \(2\)’ is one event, while ‘the die shows an odd number’ is a different event; and, in a single run, these events do occur together, if the result is \(3\) or \(5\).
1.2. Type-token distinction for random events
In the context of probability theory, we need to distinguish between types of random events and tokens of random events [15]. For that purpose, let us consider the random experiment of tossing a coin. To have a specific example, let us assume that we perform \(10\) runs and the coin shows heads in runs \(1, 2, 5, 6, 8, 9\) and the coin shows tails in runs \(3, 4, 7, 10\). Now, how many events are we talking about in this example? Are there two events, where ‘the coin shows heads’ is one event and ‘the coin shows tails’ is the other event? Or are there ten events, where we have ‘the coin shows heads in run \(1\)’ as one event, ‘the coin shows heads in run \(2\)’ as another event, ‘the coin shows tails in run \(3\)’ as yet another event, and so on?
On the one hand, we can clearly say that ‘the coin shows heads in run \(1\)’ is a different event than ‘the coin shows heads in run \(2\)’; they must be different events, as they occur in different runs. On the other hand, however, we can also clearly say that ‘the coin shows heads’ in run \(1\) is the same event as ‘the coin shows heads’ in run \(2\); they must be the same event, as ‘the coin shows heads’ in both runs. While these two statements seem contradictory, they can both be simultaneously true, because of the “type-token ambiguity” [16] of the random-event concept. When we speak of “the coin shows heads” as a random event, we mean a type. In contrast, when we speak of “the coin shows heads in run \(2\)” as a random event, we mean a token—that is, one specific instance—of the corresponding type. So, ‘the coin shows heads in run \(1\)’ and ‘the coin shows heads in run \(2\)’ are two different tokens of one random-event type.
In an actual run of a random experiment only random-event tokens can occur [17]. Note, however, that even in a set of runs [18], any random-event token can occur at most once [19]; for example, the random-event token “the coin shows heads in run \(2\)” can only occur in run \(2\) but not in any earlier or later run. Thus, in a set of runs of a random experiment, it does not make much sense to count the number of a random-event token. We can count, however, how many tokens of a random-event type occur in a set of runs. In the above coin-tossing example of ten runs, six tokens occur of the random-event type ‘the coin shows heads,’ and four tokens occur of the random-event type ‘the coin shows tails.’ For brevity, we also say that a random-event type occurred in some number of runs, which means that in this number of runs a corresponding random-event token occurred.
While the type-token distinction of random events remains relevant, it is usually clear from context whether we mean a random-event token or a random-event type. Thus, in the following, we will no longer explicitly address the type-token distinction and simply speak of random events again; unless we need to stress whether we speak of a type or a token. Having clarified the difference between types and tokens of random events, we can now turn our attention to the frequency and relative frequency of the occurrence of random events in random experiments [20].
1.3. Frequency and relative frequency
By definition, we do not know in advance whether or not a random event will occur in a run of a random experiment. That is, we cannot tell in advance whether or not a specific random-event token will occur. We can still know, however, how frequently a random-event type will approximately occur in a sufficiently large set of runs [23]. For example, we do not know in advance whether a coin will show heads for any individual coin toss, but we might still know that, for a sufficiently large number of coin tosses, heads will approximately show up about half the time.
The number of runs \(n_E\) in which a random-event type \(E\) occurred is called the frequency of the random event. When put in relation to the total number of runs \(n_r\) in which the random event could have occurred, it is called relative frequency \(\nu_E = n_E/n_r\) [24]. In the above coin-toss example with \(10\) runs, the coin shows heads \(6\) times and tails \(4\) times. Thus, the frequency of the heads event type \(H\) is \(n_H = 6\), while the frequency of the tails event type \(T\) is \(n_T = 4\). In relation to the total number of runs \(n_r = 10\), we find the relative frequencies of \(\nu_H = n_H/n_r = 6/10 = 0.6\) for heads and \(\nu_T = n_T/n_r = 4/10 = 0.4\) for tails.
Based on the concept of relative frequency, we can now define an important type of random experiments. A random experiment is called regular, if “the relative frequency of an event is approximately the same on each occasion that a set of trials is performed” [25]. These regular random experiments are the type of experiments that we are usually interested in when using probabilities.
2. From relative frequency to probability
Being able to determine relative frequencies of random events in random experiments, we might now wonder how we could predict relative frequencies in theory.
A relative frequency of a random event is specific for a set of runs of a random experiment; it might be—and typically will be—slightly different for a different set of runs. Thus, a relative frequency is an experimental or empirical quantity; not a theoretical or mathematical quantity [26]. So, in theory, we cannot proceed in the same way and count how often an event type occurs in a set of runs of a random experiment; there simply are no experiments in theory [27]. What we need instead is a mathematical theory—a theory of probability—with which we can express theoretical ‘predictions’ about experimental relative frequencies.
Tailored to the description of relative frequencies, the theory of probability should respect their general properties. Thus, to motivate the mathematical axioms of the theory of probability, we first consider a few general properties that relative frequencies have to have; we roughly follow reference [24].
2.1. General properties of relative frequencies
The frequency \(n_E\) of some event \(E\) cannot be negative, \(n_E \geq 0\), because we are only counting up if a token to the type \(E\) occurs in a run of the random experiment. This frequency also cannot be larger than the total number of runs \(n_E \leq n_r\), as there can be at most one token to the event type \(E\) per run. In combination, we find that the frequency \(n_E\) is bounded by \(0\) from below and by the total number of runs \(n_r\) from above; that is, \(0 \leq n_E \leq n_r\). Now, dividing by the total number of runs \(n_r\) [28], we find \[ 0 \leq \nu_E \leq 1\ , \tag{1} \] where \(\nu_E = n_E/n_r\) is the relative frequency of the event \(E\). Thus, the relative frequency \(\nu_E\) of any event \(E\) must lie in the interval from \(0\) to \(1\).
A certain event \(E_c\) is known to occur in advance in every run of a ‘random’ experiment; otherwise, it would not be a “certain” event. Thus, the frequency of a certain event \(n_{E_c}\) must be as large as the number of runs \(n_r\); that is, \(n_{E_c} = n_r\). Thus, the relative frequency \(\nu_{E_c} = n_{E_c}/n_r\) of a certain event \(E_c\) must always be given by \[\nu_{E_c}=1\ . \tag{2} \] Note that the reverse is not necessarily true; just because we find the relative frequency of an event to be equal to \(1\) in some set of runs, does not mean that the event is certain, which would mean that its relative frequency has to be equal to \(1\) in every set of runs.
The frequency \(n_{E_1 \vee E_2}\) of an event \(E_1 \vee E_2\) that is a logical disjunction of two mutually exclusive events \(E_1\) and \(E_2\) is given by the sum of the frequencies of these two events, \(n_{E_1} + n_{E_2}\); if event \(E_1\) occurs in \(n_{E_1}\) runs and event \(E_2\) occurs in \(n_{E_2}\) runs, then in \(n_{E_1} + n_{E_2}\) runs either \(E_1\) or \(E_2\), that is, \(E_1 \vee E_2\) occurs. Thus, the relative frequency \(\nu_{E_1 \vee E_2}= n_{E_1 \vee E_2}/n_r\) of an event \(E_1 \vee E_2\) is always given by \[\nu_{E_1 \vee E_2} = \nu_{E_1} + \nu_{E_2}\ , \tag{3} \] where \(\nu_{E_1}\) and \(\nu_{E_2}\) are the relative frequencies of the two mutually exclusive events \(E_1\) and \(E_2\) respectively.
2.2. Event space and probability axioms
We will use the general properties of relative frequencies, condensed in equations (\(1\)), (\(2\)), and (\(3\)), to motivate axioms for a mathematical theory of probability. However, before we can even express the axioms, we need to introduce the concept of an event space.
The event space \(S\) is a mathematical set whose elements can be used to represent the possible random-event types of a random experiment. More specifically, the event space has to have some internal structure that can represent logical relations between events [29]. For example, this internal structure must be able to represent the logical relation between an event \(E_1 \vee E_2\) that is a logical disjunction of two other events \(E_1, E_2\) by some mathematical relation between those three elements of the event space that can represent these three events \(E_1 \vee E_2, E_1, E_2\). It it important to note that types of random events must not be confused with elements of a mathematical set; while random events and their types have been explicitly defined above, mathematical sets and elements are implicitly defined by the axioms of set theory. So, strictly speaking, neither random events nor their types are elements of an event space. However, by construction of the event space, its elements can mathematically represent random-event types. And, because the internal structure of the event space reflects the logical relations between events, we can use the same symbols for elements of the event space as for the events they represent [30]. Even though the internal structure is important [31], we will not discuss further details here, because—being closely related to the difference between ‘classical’ and ‘quantum’ probabilities—the details are worth a separate discussion. In the following, all we need is that the event space is a mathematical set that has an appropriate internal structure to represent events and the logical relations between them.
Motivated by the general properties of relative frequencies, we now introduce a probability function \(P\) that associates a real number to every element of an event space and satisfies the following three axioms:
- (I) for every element \(E\) of the event space \(S\), we have \[ 0 \leq P(E) \leq 1\ ; \tag{4} \]
- (II) for every element \(E_c\) of the event space \(S\) that can represent a certain event [32], we have \[ P(E_c) = 1\ ; \tag{5} \]
- (III) for every element \(E_1 \vee E_2\) of the event space \(S\) that can represent an event that is a logical disjunction of two mutually exclusive events represented by the two elements \(E_1, E_2\) of the event space, we have \[ P(E_1 \vee E_2) = P(E_1) + P(E_2)\ . \tag{6} \]
These three axioms are sufficient to construct an elementary theory of probability [24].
Before we can finally define what a probability is, we should note two important points. First, the above axioms of the theory of probability are motivated by—but not derived from—the general properties of relative frequencies. Second, the event space is a mathematical set with an appropriate internal structure to allow us to represent the logical relation between events; this mathematical set and its elements do not have to actually represent events and logical relations between them. So, despite its name, the event space does not have to represent any events of a random experiment; instead, it is just a mathematical set with some specific internal structure.
2.3. What is a probability?
Finally, we can give a clear answer to the question “What is a probability?”
Every function that associates a real number to every element of an event space and fulfills the above axioms is a probability function; and the corresponding function values are the probabilities. In other words, a probability is a real number associated by a probability function to an element of an event space.
At first glance, this answer might seem to be quite underwhelming: a probability is just a real number associated by a mathematical function to an element of a mathematical set. But the mathematical set—the event space—is not an arbitrary mathematical set; it carries an internal structure that allows us to represent events and their logical relations. And the mathematical function—the probability function—is not an arbitrary mathematical function; it fulfills axioms that make it suitable to describe relative frequencies in random experiments. So, as a mathematical concept, probability might not be too interesting in and of itself; however, the mathematical concept of probability is interesting because of its scientific applications.
Conclusion
In this article, we have motivated the axioms of an elementary theory of probability from basic properties of relative frequencies. However, the resulting concept of probability, which we defined as a purely mathematical concept, is independent of this motivation. Giving a purely mathematical definition of probability, we are able to clearly state what a probability is: it is a number that is associated by a probability function—a function that satisfies the axioms of probability theory—to an element of an event space.
The benefit of the purely mathematical definition is that it allows us to sidestep the debate around the meaning of probability; there simply is no meaning beyond the mathematical definition. What then happens to the debate? The debate around the meaning of probability turns into a debate about applications of the probability concept. However, that basically resolves a key issue of the debate: while there should be, arguably, only one meaning of the probability concept, there is no problem in having multiple applications of it.
One key application of the probability concept is the description of relative frequencies of random events in random experiments. Though, one might wonder: what is the advantage of the probability concept over the relative-frequency concept? A relative frequency of a random event is an experimental quantity that is associated to an actual set of runs of a random experiment. In contrast, a probability of a random event [33] is a theoretical quantity that can be associated to the random experiment itself. So, while a relative frequency can—and typically will—change from one set of runs to another set of runs, a probability remains invariant across different sets of runs as long as the random experiment remains the same. The relation between probability and relative frequency is generally complicated and falls into the realm of statistics. Roughly speaking, however, in a sufficiently large set of runs of a random experiment, the relative frequency of an event should approach the probability associated to that event.
Based on the clear distinction between the mathematical concept of probability and the experimental concept of relative frequency, we can also understand the difference between a priori and a posteriori probabilities for this context. We speak of a posteriori probability, if we estimate the probabilities of random events in a random experiment from the relative frequencies we found in a set of runs, which is possible only after we run the random experiment several times. For example, when we toss a coin \(1000\) times, and find heads \(H\) in \(509\) runs and tails \(T\) in \(491\) runs we might simply estimate the probabilities by \(p_H \approx 0.509 = \nu_H\) for heads and \(p_T \approx 0.491 = \nu_T\) for tails [34]. In contrast, we speak of a priori probability, when we obtain the probabilities of the random events in a random experiment based on theoretical principles, which is possible even before we even run the random experiment for the first time. For example, in a coin toss, because of the approximate symmetry between heads \(H\) and tails \(T\), we might reasonably assume the corresponding probabilities to be approximately equal \(p_T \approx p_H\) even before we toss the coin for the first time. Such an a priori probability statement can be tested empirically by running the random experiment sufficiently often [1, 37] and using a statistical test [38] to check whether the experimentally obtained relative frequencies fit reasonably well to the theoretically assumed or predicted probabilities.
The statistical description or prediction of relative frequencies in sets of runs of random experiments, as we just discussed, is an essential scientific application of probability theory. Note that, in this application, we only make statements about the occurrence of random-event types in sets of runs of a random experiment. Whether or not probability theory can also be applied to make statements about the result of a single run of a random experiment, as in the sense of Popper’s propensity interpretation [2, 3], is a separate question. Similarly, whether or not the mathematical concept of probability can be applied to describe a subjective degree of believe in the truth of some propositions, as in some versions of inductive inference or Bayesianism [2, 3], is also a separate question. These questions, however, go far beyond this article. So, to conclude, we just note that a purely mathematical definition of probability can have multiple scientific applications that must be justified individually and can be used independently.
Acknowledgements
I thank Mathias Gutmann and Johannes Kleiner for fruitful discussions.
License and citation.
This work was published by Tim Ludwig on 31 December 2025 in the Philosophy for Physics blog, accessible under https://timludwig.de/blog-philosophy-for-physics/, and it is openly licensed via CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). To give appropriate credit, in academic contexts, please cite as follows: Tim Ludwig, “What is a probability?” Philosophy for Physics (blog), December 31, 2025, https://timludwig.de/philosophy-for-physics-2025-december-31/.
Footnotes
[1] Mehran Kardar, Statistical Physics of Particles (2007; repr., Cambridge: Cambridge University Press, 2017), 36.
[2] Leslie E. Ballentine, Quantum Mechanics: A Modern Development (1998; repr., Singapore: World Scientific, 2010), 31-33.
[3] Jürgen Mittelstraß ed., Enzyklopädie Philosophie und Wissenschaftstheorie, 2nd ed., vol. 8: Th-Z (Stuttgart: Verlag J. B. Metzler, 2018), s.v. “Wahrscheinlichkeit.”
[4] Friedrich Kirchner, Carl Michaëlis, and Johannes Hoffmeister, Wörterbuch der philosophischen Begriffe, ed. Arnim Regenbogen and Uwe Meyer (2013; repr., Hamburg: Felix Meiner Verlag, 2020), s.v. “Wahrscheinlichkeit.”
[5] Crispin Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, 4th ed. (Berlin: Springer, 2009), 25.
[6] It is possible to avoid the cumbersome phrasing after introducing the internal mathematical structure of the, so called, event space. Thus, the cumbersome phrasing is most certainly not the only reason for the usual lack of separation between the mathematical concept of probability and its scientific applications.
[7] A. N. Kolmogorov, “The Theory of Probability,” in Mathematics: Its Contents, Methods and Meaning, ed. A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev, and trans. ed. S. H. Gould (1969; repr., Mineola, NY: Dover Publications, 1999), 229.
[8] Kirchner, Michaëlis, and Hoffmeister, Wörterbuch der philosophischen Begriffe, s.v. “Zufall.”
[9] Here, we assumed that it is part of the complex conditions that nothing else is blocking the book’s path to the ground. If it was not known whether or not the book’s path to the ground is blocked, the event ‘the book hits the ground’ would be a random event, because we would not know in advance whether or not it will occur.
[10] Kolmogorov, “The Theory of Probability,” 230.
[11] Günter Vojta and Matthias Vojta, Teubner-Taschenbuch der statistischen Physik (Stuttgart: B. G. Teubner, 2000), 27.
[12] K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering, 3rd ed. (2006; repr., Cambridge: Cambridge University Press, 2024), 1119.
[13] F. M. Dekking, C. Kraaikamp, H. P. Lopuhaä, and L. E. Meester, A Modern Introduction to Probability and Statistics: Understanding Why and How ([London?]: Springer, 2005), 13.
[14] Josef Honerkamp, Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis, trans. Katja Lindenberg (New York: VCH, 1994), 8.
[15] In some contexts, it would be beneficial to be more precise and further distinguish between an event, the occurrence of an event, and the proposition that an event has occurred. However, for this article not much would be gained by keeping up this distinction; thus, for simplicity, we will not make it.
[16] Simon Blackburn, The Oxford Dictionary of Philosophy, 3rd ed. (Oxford: Oxford University Press, 2016), s.v. “type-token ambiguity.”
[17] Note that, strictly speaking, a random-event type is not an event, because it cannot occur. When we say that a random-event type occurs for brevity, it should be understood as a random-event token to that random-event type occurs.
[18] Note that “set” in the expression “set of runs” should not be understood in the sense of a mathematical set; the ‘elements’ of a set of runs—that is, the runs of a random experiment—are not mathematical objects. Instead, one should understand “set” in “set of runs” in the non-mathematical sense of a ‘collection’ or a ‘series’ or something similar.
[19] One might argue that “at most once” does not make much sense, because if the random-event token would ‘occur zero times’—that is, if it would not occur at all—there would be no token to the corresponding type. Thus, one could only say that a random-event token occurs exactly once or not at all, which—strictly speaking—is different from it occurring one or zero times. However, in the following, this difference will not be relevant, such that we will disregard it.
[20] Note that, similar to random events, we could also distinguish between types and tokens of random experiments. However, that distinction is already implicitly made when we speak of a run of a random experiment (token) in contrast to the random experiment itself (type). Thus, it is not necessary to introduce the type-token distinction explicitly for random experiments. It would make sense, however, to introduce a type-token-thing distinction or, more precisely, a concept-object-thing distinction, as developed in reference [21] and, in more detail, in reference [22]. What is the very same coin toss on the level of things can be analyzed on the conceptual level in completely different ways; for example, once with the coin as a classical object whose motion we can predict, and once with the coin toss as a simple random experiment with two outcomes (heads or tails). Thus, the concept-object-thing distinction of random experiments is related to the concept of relative randomness and, in turn, also to difference between relative randomness and absolute randomness; while worth a separate discussion, this difference is beyond the scope of this article.
[21] Mathias Gutmann, Leben und Form: Zur technischen Form des Wissens vom Lebendigen (Wiesbaden, DE: Springer VS, 2017), pt. 1.
[22] Mathias Gutmann, Anthropologie des Erkennens: Dialektische Studie zum Verstehen, Erklären und Begreifen (in print).
[23] Because we discuss random events and how frequently they occur in random experiments only as a motivation for probability theory, in this article, we can rely on a rough understanding of “approximately” and “sufficiently large.” To give a precise meaning to these terms, we would need to resort to statistical tests and estimators, which goes beyond this article.
[24] Kolmogorov, “The Theory of Probability,” 229-233.
[25] Riley, Hobson, and Bence, Mathematical Methods for Physics and Engineering, 1124.
[26] Of course, a frequency and the total number of runs are natural numbers. And a relative frequency, in turn, is a rational number. So, in the sense of being numbers, frequency and relative frequency are mathematical. However, they are not mathematical in the sense of being part of some mathematical theory for frequency or relative frequency. In other words, frequency and relative frequency are mathematical as numbers; but not as frequency or relative frequency. That is, there is a difference between the natural number \(5\) and the frequency of a random event in a set of runs of a random experiment being \(5\).
[27] One might argue that a simulation of a random experiment on a computer is a theoretical experiment. However, that would presuppose the concept of probability and, with it, the knowledge about a theoretical description of relative frequencies. Thus, even if one considers such simulations of random experiments to be theoretical experiments, one could not use them as a basis to develop a theory of probability.
[28] Note that we can assume the total number of runs to be positive \(n_r >0\); otherwise, without any runs (\(n_r = 0\)), we would have nothing to analyze anyway.
[29] Heinz-Peter Breuer and Francesco Petruccione, The Theory of Open Quantum Systems (2002; repr., Oxford: Oxford University Press, 2010), 3-4.
[30] Note that using the same symbols is possible, but only because of the combination between the one-to-one correspondence of events and elements and the one-to-one correspondence between the related structures; here, “related structures” means the logical relation between the events on the one hand and the internal mathematical structure of the event space on the other hand.
[31] The internal structure of the event space is not only important to avoid contradictions [14] but also, as we will see, to express the axioms of probability theory.
[32] A certain event is given by the logical disjunction \(E_1 \vee E_2 \vee E_3 \vee …\) of all elementary events \(E_1, E_2, E_3, …\), because there will occur one outcome as a result for each run of a random experiment.
[33] When we speak of a “probability of a random event,” we mean the probability that is associated to the element of the event space that represents that random event.
[34] Such a basic estimate of probabilities can be made in the spirit of reference [24]. For more elaborate estimates of probabilities, see references [35] and [36] for example.
[35] Kevin Cahill, Physical Mathematics (2013; repr., Cambridge: Cambridge University Press, 2014), 543-550.
[36] Josef Honerkamp, Statistical Physics: An Advanced Approach with Applications, 3rd ed. (Berlin: Springer, 2012), ch. 8.
[37] Ballentine, Quantum Mechanics, 46.
[38] For details on statistical tests, see references [39] and [40] for example.
[39] Cahill, Physical Mathematics, 551-560.
[40] Honerkamp, Statistical Physics, ch. 13.