Probability is defined in mathematics in the context of discrete elements in sets. However, it can be transitioned into an analogous definition in continuous mathematics. Thirdly, it can be represented as a meld of discrete and continuous concepts as a continuous probability function.
The Discrete Context
In discrete mathematics, probability is the fractional concentration of an element in a logical set. It is the ratio of the quantity of elements of the same ID to the total number of elements in the set. If the numerator is zero, the probability is zero. The probability is never negative because the numerator is never negative and the denominator is a minimum of one. Probability reaches a maximum of one, where the set of elements is homogeneous in ID. Probability can have any value from one to zero, because the denominator can be increased to any positive integer. Thus, probability in its discrete definition is itself a continuous variable, having a range of zero to one.
A probability and its improbability are complements of one.
The Continuous Context
Probability is a fraction of a whole set of discrete elements. If, however, we define a whole as continuous, we can then define probability in a continuous context analogous to its definition in a discrete context. One example is had in identifying the area of a circle as a continuous whole and identifying segments of area using different IDs, such as in a pie chart. Another example is that in statistics where the whole is defined as the area under the normal curve. Probability is then a fraction of the area under the curve.
Melding the Discrete and Continuous as a Probability Function
The simplest set of discrete elements is that in which all the elements have the same ID. The next simplest is that in which the elements are either of two IDs, where the quantities of the elements of each ID are equal. Thus, the probability of each element is one-half and the improbability of each is one-half.
If we choose a continuous function which oscillates between two extremes, we could associate the one extreme with an ID and the other extreme with a second ID. We could view the first ID as having a probability of one at the first extreme and having a probability of zero at the other extreme. We would thus be viewing the function as a probability, which transitions continuously through the intermediate values as it cycles between a probability of one and a probability of zero, i.e. between the two IDs.
At this second extreme, the second ID has a probability of one, which is also the improbability of the first ID.
A visual example would be a rotating line segment oscillating at a constant angular velocity between a horizontal orientation as one ID and a vertical orientation as the second ID. The continuous equation for this would be the cos^2 (α). The function oscillates from the horizontal or from a probability of one at α = 0 degrees to the vertical or to a probability of zero at α = 90 degrees. At α = 180 degrees, it is horizontal again with a probability of one. The probability goes to zero at 270 degrees and back to one at α = 360 degrees. The intermediate values of the function are transient values of probability forming the cycle from one to zero to one and back again from one to zero to one.
The improbability of horizontal, namely the probability of vertical, would be the sin^2 (α). The probability of horizontal plus its improbability equals one. Thusly, cos^2 (α) + sin^2 (α) = 1.
The Flip of a Coin
The score was tied at the end of regular time in the NFC Championship Game in 2016 between the Green Bay Packers and the Arizona Cardinals. This required a coin toss between heads and tails to determine which team would receive the ball to start the overtime. However, in the first toss, the coin didn’t rotate about its diameter. The coin didn’t flip. Therefore the coin was tossed a second time.
If, rather than visualizing a line segment, we envision a coin rotating at a constant angular velocity, we wouldn’t choose horizontal vs. vertical as the probability and improbability, because we wish to distinguish one horizontal orientation, heads, from its flipped horizontal orientation, tails.
A suitable continuous function of probability, P, oscillating between a value of one or heads at the horizontal α = 0 degrees and a value of zero, or tails at the horizontal α = 180 degrees, would be
P = [(1/2) × cos (α)] + (1/2), where the angular velocity is constant.
The probability of tails is 1 – P = (1/2) – [(1/2) × cos (α)]
The probability of heads and the probability of tails are both one-half at α = 90 degrees and α = 270 degrees.
The probability of heads plus its improbability, which is the probability of tails, is one
Whether we visualize these functions, the one as oscillating between horizontal and vertical and the other as oscillating between heads and tails, the functions are waves.
We are thus visualizing a probability function as a wave oscillating continuously between a probability of one and zero. The probability is the fraction of the maximum magnitude of the wave as a function of α.
An Unrelated Meaning of Probability
We use the word, probability, to designate our lack of certitude of the truth or falsity of a proposition. This meaning of probability, reflects the quality of our human judgment, designating that judgment as personal opinion, rather than a judgment of certitude of the truth. This meaning of probability has nothing to do with mathematical probability, which is the fraction of an element in a logical set or, by extension, the fraction of a continuous whole.
Driven by our love of quantification, we frequently characterize our personal opinion as a fraction of certitude. This, however, itself is a personal or subjective judgment. A common error is to mistake this quantitative description of personal opinion to be the fractional concentration of an element in a mathematical set.
Errors Arising within Material Analogies of Probability
A common error is to identify material analogies or simulations of the mathematics of probability as characteristic of the material entities employed in the analogies. In the mathematics of probability and randomness the IDs of the elements are purely nominal, i.e. they are purely arbitrary. The probability relationships of a set of six elements consisting of three elephants, two saxophones and one electron are identical to those of a set of three watermelons, two paperclips and one marble. This is so because the IDs are purely nominal with respect to the relationships of probability.
In analogy, the purely logical concepts of random mutation and probability are not properties inherent in material entities such as watermelons and snowflakes. This is in contrast to measureable properties, which, as the subject of science, are inherent in material entities.
The jargon employed in analogies of the mathematical concepts also leads us to confuse logical relationships among mathematical concepts with the properties of material entities. In the roll of dice we say the probability of the outcome of boxcars is 1/36. We think of the result of the roll as a material event, which becomes a probability of one or zero after the roll, while it was a probability of 1/36 prior to the roll of the dice. In fact, the outcome of the roll had nothing to do with probability and everything to do with the forces to which the dice were subjected in being rolled. The analogy to mathematical probability is just that, a visual simulation of purely logical relationships.
We are also tempted to think of the probability 1/36 as the potential of boxcars to come into existence, which after the roll is now in existence at an actuality of one, or non-existent as a probability of zero. In this, we confuse thought with reality. Probability relationships are solely logical relationships among purely logical elements designated by nominal IDs. Material relationships are those among real entities, whose natures determine their properties as potential and as in act.
In quantum mechanics, it is useful to treat energy as continuous waves in some instances and as discrete quanta in others. It is useful to view the wave as a probability function and the detection or lack of detection of a quantum of energy as the probability function’s collapse into a probability of one or of zero, respectively.
As an aid to illustrate the relationship of a probability function as a wave and its outcome as one or zero in quantum mechanics, Physicist, Stephen Barr, proposed the analogy,
“This is where the problem begins. It is a paradoxical (but entirely logical) fact that a probability only makes sense if it is the probability of something definite. For example, to say that Jane has a 70% chance of passing the French exam only means something if at some point she takes the exam and gets a definite grade. At that point, the probability of her passing no longer remains 70%, but suddenly jumps to 100% (if she passes) or 0% (if she fails). In other words, probabilities of events that lie in between 0 and 100% must at some point jump to 0 or 100% or else they meant nothing in the first place.”
Problems with the Illustration
The illustration fails to distinguish the purely logical relationships of mathematical probability from the existential relationships among the measurable properties of material entities. The illustration identifies probabilities as being of events rather than identifying probabilities as logical relationships among purely logical entities designated by nominal IDs. It claims that probability must transition from potency to act or it is undefined. In contrast, probability is the fractional concentration of an element in a logical set. The definition has nothing to do with real entities, whose natures have potency and are expressed in act.
Another fault of the illustration is that it is not an illustration of mathematical probability, but an illustration of probability in the sense of personal opinion. Some unidentified individual is of the opinion that Jane will probably pass the French exam. The unidentified individual lacks human certitude of the truth of the proposition that Jane will pass and uses a tag of 70% to express his personal opinion in a more colorful manner.
It is a serious error to pick an example of personal opinion to illustrate a wave function, viewed as a probability function. A wave function, such as that associated with the flip of a coin oscillating between heads as a probability of one and tails as a probability of zero, would have served the purpose well.
Of course, a wave, viewed as a probability function, is not the probability of an event. It is the continuous variable, probability, whose value oscillates between one and zero, and as such assumes these and the intermediate values of probability transiently. The additional condition is that when the oscillation is arrested, the wave collapses to either of the discrete values, one and zero, the presence or absence of a quantum. The collapse is the transition of logical state from one of continuity to one of discreteness.