Philosophical Discussion

The Probability Laws of Parapsychology

by Robert G. Howard, Ph.D.

Abstract: J. B. Rhine and others asked, “What would lead to general scientific attention to parapsychology?” This article and previous articles have defined the general theory in their answer, “A general theory that would harmonize psi communication with the known energetic system of the universe.” Rhine was one of the first researchers to use probability mathematics to gain scientific attention to parapsychology. This article suggests many more tools in the mathematics of probabilities. Three hypotheses are proposed to connect the physical part of parapsychology to the psychical part through probability mathematics.


There is a bias in scientists to believe that cause yields effect and that a given event was caused by a well determined set of causes. A precise description of cause and effect relationships are required in order to get a solution to the math of differential equations and calculus. The development of these types of math required an exact knowledge of initial conditions and boundary conditions. About 1760, a few Europeans began the math of probabilities which allows for unknown relations between cause and effect and for unpredictable events. This led to the recognition that there are effects with unknown initial factors, unknown causes, or a limited set of known influences.

Probability mathematics requires the admission that much of humans’ experience is mysterious and even unknowable. The premise of probability laws recognizes that not all causes and effects could be known. So there was the progression of consensus to believe in cause-effect then partially known cause -partially known effect, then unknowable cause- unknowable effect. This suggests that there are epistemological processes of knowing or acquiring knowledge that humans do not know or allow themselves to ponder or to believe. In parapsychology research, toleration for the unknown is helpful.

Probability laws were suggested first by Richet and Edgeworth for use in psychical research to detect the effect of volition on extrasensory perception (ESP) ( Richet, 1884) and (Edgeworth, 1885). Psychic science grew into the more precise and rigorous science of parapsychology when the laws of probability were applied, for example by Rhine at Duke University (Pratt, Rhine, Smith, Stuart, Greenwood, 1966, p. 22ff). They presented “ the Mathematical Methods” to prove that extrasensory phenomena are reproducible and experimentally controllable due to cause - effect connections and thus are not random events.

Rhine and others asked, “What would lead to general scientific attention to parapsychology?” They gave two answers, “ Easy application in daily use and a general theory that would harmonize psi communication with the known energetic system of the universe.” (Pratt, Rhine, Smith, Stuart, Greenwood, 1966, p. iv) Applications in daily use will be addressed in another paper. This article and previous articles have begun defining the general theory that would harmonize psi communication with unknown psychic systems which Rhine and others requested (Howard & Kelleher, 2008b, 2008c, and 2009b).

This article suggests many more tools in the mathematics of probabilities for giving credibility to this science. These heuristics may stimulate other researchers to employ them to analyze psychic data in more depth. This article also introduces concepts that are the foundation for two more articles that follow in another issue of this journal; articles on the diffusion of extra sensory data into the conscious psyche, and on the dependence of extra sensory perception on the third dimension of time. The third dimension of time connects realities through probabilities, not cause and effect.

This is the time to discover the mathematical relationships and laws that exist in nature, which govern psychic science. Physical laws were discovered thru observation, intuition and the desire to order nature. The laws of nature encourage falsifiable predictions. Some sciences, like psychic science, do not have well formulated laws. Various mathematical structures are offered below which explain, on general grounds, why observed psychic phenomena occur. They are building blocks for greater mental constructions later. The laws of probability predict quantitative relations between measurements few people would have anticipated. The study of the connection between the physical base of parapsychology and the psychic structure of nature is uncovering a hidden structure of which we humans are part.

This article is addressed to non-scientists, assumes no scientific background but uses high school algebra. One objective is to improve scientific innumeracy by teaching how to extend the quantitative reasoning about the uses of probability in parapsychology.

The Likely the Unlikely, and the Incomprehensible

The most basic concepts in probability mathematics are briefly listed below to convince the Reader that this is tractable mathematics. See Appendix. Probability of an event is rated zero for no event and one for total probability, certainty of the event. The concept of probability requires a repeatable experiment of observation with more than one possible outcome controlled by chance. Which means that before the fact, precisely which outcome will occur is neither known nor deducible. For this experiment, a probability of a given outcome is a numerical estimate, based on experience or theory, of the fractional occurrence of that outcome in a large number of trials. Probability laws do not yield good distributions with a small number of trials. Probabilistic laws will be applied in more penetrating ways below.

How are the probabilities of complicated outcomes deduced from those for elementary ones? This is answered by math rules.

Rule (1) The probabilities for all mutually exclusive outcomes add up to one. This sum is equal to the probability that an arbitrary one of the mutually exclusive outcomes will occur. Rule (2) The probability for two independent events both to occur is the product of the probabilities for the separate events.

Rule (3) Even when two events are not independent, the probability of both occurring is the probability of the first event times the probability of the second, subject to the condition that the first has occurred.

Probability of Normality and Large Numbers

An experiment on psychic phenomena with random outcomes performed a large number of times yields a probable fact of nature. The fact is that only a very small number of probable outcomes associated with many trials has a probability of occurring. This mathematical phenomenon leads to near certainty from the chaos of uncertainty in many important cases in physics and in psychic phenomena.

The Probable Binomial Distribution

To show how easy to use the math tools are, there follows more brief summaries of elementary concepts, and terminology, in probability mathematics. The binomial distribution considers an experiment with two outcomes, but the math can be expanded to any number of outcomes. To review the missing steps or to use the mathematics, see Reasoning about Luck: Probability and Its Uses in Physics (Ambegaokar, 1996, p. 23ff).or any elementary probability book.

Consider an experiment which consists of N trials of a 2 outcome experiment (like tossing a six sided die). One outcome is when a six shows on the die; success with probability p.= success. Another outcome is when a six does not show, failure with probability (1-p) =failure. The shorthand expression is (N:r) meaning that there are N combinations taken r at a time. What is the probability, P(r) of r successes in N trials? It is based on 3 factors

1. The number of ways in which the r successes can occur during the N trials, which is precisely the number of combinations of N things taken r at a time, for example, on a die marked with one to six dots.

2. The probability of r successes, P( r ) when a six shows, is the probability of one success multiplied by itself r times

3. The probability of (N - r) failures (when a six does not show) , or equivalently, the probability of one failure multiplied by itself (N - r) times

P( r ) = (N:r) pr ( 1-p)N-r equation (1)

The collection of numbers P( r) is the probability distribution for

r = 0,1,2,3,4,…N integers

If there are N combinations taken r at a time, this is written

(N:r) = {N(N-1)…(N-r+1)}/{r(r-1)…1} = N!/{r!(N-r)! equation (2)

Equation (1) is called the binomial distribution. “bi” means there are 2 possible outcomes; for example, a die can show a six or not. Note that equation (1) gives the following.

(a.) When N increases, the lopsided distributions of a plot of P( r ) versus N, for small N, become symmetric and look like a bell.

(b.) As N increases, the distributions of a plot of P( r ) versus N, begin to peak at the same fractional distance along the N axis of the plot. For a die with six sides, the maximum occurs when r = N/6

(c.) As N increases, there is a trend toward narrower peaks. This is remarkable: The number of outcomes for which there is an appreciable probability becomes a smaller fraction of all possible outcomes as the number of trials, N increases. So the equation gives a quantitative assessment of likelihood or certainty. Certainty eludes us when we deal with random events. The statistician must construct reasonably sure tests of hypotheses from a fraction of all the potentially available data. In parapsychology, there is usually extremely large numbers of repetitions of an experiment. So one does not test a fraction. One can use these simple rules and formulas of probabilities to show that a psychic reality is not random or is extremely unlikely based on cause and effect. (Ambegaokar, 1996, p.23ff)

The Probable Mean and the Probable Standard Deviation

A complicated random process, such as many trials of a simple random experiment has many possible outcomes. The collection of probabilities is a “distribution.” If a distribution is peculiar, bumps and wiggles, the only way of conveying the information it contains is to list all its entries one column of probabilities and one column of the number of successes. A way to describe the shape of the distribution of successes and Number of trials P( r ), comes next.

The definition of “average” is mean = mu which can be written

mu = Equation (3)

The mean square deviation = variance = sigma2

And sigma2N = Equation (4)

The connection between standard deviations for trial sizes of N+1 and N and 1 is

sigma2N+1 = sigma2N + sigma2 1

And the standard deviation is sigma N = N sigma1

(Ambegaokar, 1996, p. 30ff).

The probable Normal Distribution

The normal probability distribution is called the Gaussian distribution, a curve depending on a continuous variable, X and 2 numbers called mu and sigma. The curve is bell shaped. The area under it = 1. The peak is at X=mu and the standard deviation is sigma. A region of 2.6 standard deviations on either side of the mean contains 99.07% of the area under the normal curve. The point is that after a moderate number of trials of a two outcome random experiment, an approximation of the normal distribution emerges with the same mean and standard deviations as the N trial distribution. One standard deviation on either side of the mean contains approximately 68% of the probability. The probable outcomes are distributed in a small range, which is more like a near -certainty than a wholly unknown state. Some outcomes are extremely unlikely to occur. The point made here is that the distribution of probabilities changes a wholly unknown process within the psyche into one of relative certainty. This is demonstrated in a more convincing way in Figure 8.1 in Ambegoakar (Ambegaokar, 1996, p. 120ff). The figure shows the Binomial and Normal Probability Distribution. This is a common figure in probability mathematics. Thus, it is not shown here.

The Probability of Determining Whether One is Perceiving Without the Normal Five Senses

Assuming that both the unconscious and the conscious psyche are receiving extra-sensory data continuously, how does one identify the extra-sensory data from the sensory data? The various mathematical probabilities calculated below describe various difficulties of becoming aware of psychical information. The states of mind are the results of probable, not certain, states of the psyche. For example, sense data and extrasensory data are not well determined, but are probable. The states of mind approximate synchronicity, more than cause and effect connections (Jung, 1955, p.1-146 ). Pauli discussed the inadequacy of the “deterministic” philosophy and suggested probability mathematics (Pauli, 1955, p.147-233). Could one consider that a synchronistic connection can be assigned a probabilistic description?

There is a long list of problems to solve in order to identify the extra-sensory data; one is loss of memory. Precognition during dreams is a form of sensing information from the unconscious psyche, and also from extra-sensory sources. Dream contents are registered in short term memory. They are lost if not transferred to long term memory immediately upon waking. The following is the mathematical description of this loss of memory.

Conservation Law for Psyche Contents

Loss of memory of an extra-sensory event is one reason why more people do not experience such an event. How is loss of memory defined as a probable event? First, ponder the conservation of the contents of the conscious or unconscious psyche in terms of contents increasing. Second, define the probability of memory loss. In physical transport theory, which treats the diffusion of a working substance through a medium, there is a description of rate of storage of any entity, such as psyche contents, within a volume, V. We can use this principle for a duration of time short enough that there are small changes in contents.

Assuming that someday there will be a way of measuring unconscious content (UCC), and quantifying the rate of UCC storage in memory in psyche volume, V , it would take the following form

{UCC/t} = d/dt Equation (5)

Equation (5) is the form because it describes any phenomenon with the properties described above. The basic mathematics for storage in equation (5) are taken from Unified Analysis and Solutions of Heat and Mass Diffusion (Mikhailov and Ozisik, 1984, p. 1-30 )

Probable Decay of Psyche Contents or Loss of Memory

Equation (5) will be negative storage, loss of memory of the UCC, if the second term in equation (5) is zero. This is the leaking of UCC out of the psyche; the loss of a memory. It could also be the transmutation of a UCC memory into a different UCC.

The math of decay of psyche, equation (5) , is a mental gateway into the unknown topic of loss of ESP or the lack of awareness of ESP.

The same decay process always occurs at different durations of time after a given initial time. The probability P of decay, in any small interval of time is proportional to the interval. Divide a given time t by M intervals where M is chosen sufficiently large that t/M is small.

So P(t) = Gdecayt/M

where the proportionality constant is Gdecay, the decay rate. And the probability of

no decay during t/M is 1-P(t)

The probability NP(t), of a memory that does not decay during t/M is

thus NP(t) = [1 - Gdecayt/M ]M

If M is much larger than t, then NP(t) = exp(-Gdecayt)

Thus there is a mathematical description of losing memories of ESP that is received in the unconscious contents and not losing such ESP memories. The math is the heuristic of describing the loss ESP or the lack of awareness of ESP.

Probability Laws of Quantum Theory Describe Some Neural Processes

The neurons have dimensions on the order of atomic concepts. Therefore quantum theory is applied to the decay phenomenon. Quantum mechanics can only predict probabilities in spite of knowing the starting conditions, even when the initial conditions are specified as precisely as possible. The laws of probabilities describe the process.

Three Hypotheses About Combined Physical and Psychical Concepts Based on Probability

Parapsychology describes non-physical reality, which is based in physical reality. Mental constructs of non-physical reality are enfolded within other mental constructs such as physical reality and are dependent on them. This is described mathematically by the theory of functions. The following constructs are functions of both physical and psychic dependent variables. The concept of prana (which can be translated, ZEST), equilibrium, entropy, and disorder are used in computing the probability laws of UCC and conscious content (Ambegaokar, 1996, p.110ff). Each variable may be defined as dependent on defined or undefined, known or unknown variables. This is a commonly accepted practice in all the sciences.

Prana Hypothesis

The ancient Indians coined the word, prana. This could be a concept similar to, “ ZEST, ” total motive power, elan vital, vitality, or libido. I propose the hypothesis that prana is an element associated to each conscious content and UCC:

Prana = F(EV) + G(E) equation (7)

Where F(EV) is some math function, F, depending on elan vital(EV) which is associated with each UCC component. Each UCC is built up from many component memories in the neurons and the psyche. EV is generated by the psychic field and is part of the UCC diffusion process.

This energy, G(E), can be deduced from the Hodgkin-Huxley equations defined in “A quantitative description of membrane current and its application and excitation in nerve” (Hodgkin, Huxley, 1952). More background for transport of contents within the brain is in Alwyn (Alwyn, 1995, Appendices F &G).

Elan vital is analogous to the energy described by kBT, the Boltzmann energy description of temperature.

G(E) is some mathematical function, G, depending on the physical energy, E, of the neural matrix. E may be defined, for example, in the Hodgkin-Huxley equations which describe the transfer of a voltage difference and a permeability of a neuron. See the tutorials, “Introduction to Neurons” with four fundamental references on the theory and mathematics of neuro-science (Friedman, 2005, p. 1-20) and “An Introduction to Dynamical Systems and Neuronal Dynamics” which includes an 46 references (Terman, 2005, p. 21-68)

Physical energy transfer is the analogy to the psychical elan vital transfer.

Prana does the work of diffusing associated UCC across the barrier into the conscious psyche, of absorbing EV and E, and of giving up the EV and E at the end of the diffusion cycle. Thus the psyche contents include prana which is the combination of psychical components, physical components. Prana is calculated as a probability of a large number of UCC components.

Now define other constructs of the psyche.

Equilibrium = is the condition in which the UCC components are stirred up (disorganized) as much as possible

Entropy = a way of quantifying equilibrium by using prana

Maximum entropy = depends on the constraint of fixed average prana as a way of deducing the probability that a component will be found in a particular one of its states of diffusion

Psyche Equilibrium as a Time Average of States Hypothesis

Preliminary to describing the transport of psyche contents, define an equilibrium psyche state.

The molecules of a gas are analogous to components of a mental construct. UCC are analogous to the working substance in a physical diffusion of gas.

Equilibrium is the condition when UCC are in a state which does not change during a relatively short time; no change as far as averages measured over a short duration of time over many components are concerned. No UCC components enter or leave or are changed within the volume, V, of the psyche.

Diffusion of Unconscious Contents Hypothesis

Assume that it is possible to define at least one UCC in terms of individual neural components plus psychic components. Crick’s research in The Astonishing Hypothesis: The Scientific Search for the Soul , suggested that any given perception is the result of many components pulled from various parts of the neural matrix where they were stored previous to the recall to perception (Crick, 1994).

Assume this one UCC are made up of components of which the average prana is much larger than the average prana of other UCC. Assume that diffusion of the UCC is described by known psychic laws (Howard &Kelleher, 2009b) and by known physical laws such as the Hodgkin-Huxley equations.

Probability Mathematical Description of Entropy Associated with Disorder in the Psyche

How does order in the psyche affect perception of extrasensory data? Under what condition does one perceive extra sensory data originating outside the psyche? For the purpose becoming conscious of data not originated in the senses, or the psyche, is it better to have an ordered psyche or a completely disordered psyche? Another way of asking these questions, “Does extrasensory data diffuse into the conscious from a psyche in a disordered condition?”

Disorders of the psyche have been studied for over 100 years. Can they be described mathematically as a distribution of components cited in Crick, Terman, and Friedman above? How to find a number to represent the randomness, the worst type of disorder, in a distribution of UCC components? What is the non-equilibrium probability of entropy and disorder in the psyche? What is the description of psyche components at a little less than equilibrium, what is the least ordered, and what components are completely ordered? The answers will be presented in another paper.

Maximizing Entropy and Minimizing Order at Equilibrium

Use these heuristic tools to seek the probability distribution for a set of numbers 0< n <1, which sum to 1.0 and maximizes entropy, with a single number, constrained by conservation of prana and UCC. Figure 1 shows such a distribution. The outcomes that the probabilities refer to, are the states of diffusion of components of a large UCC. Time is not a parameter. The result is a unique time -independent distribution describing equilibrium and gives a precise and insightful meaning to the concept of prana. This is an example of how probability math provides much from very little.

Figure 1 Entropy between zero and one

Definition of Entropy in the Psyche

Entropy has meaning in neuroscience and parapsychology because order and disorder have meaning. Information diffuses thru the psyche and becomes disordered. Maximum disorder relates to diffusion of unconscious contents across the barrier into the conscious psyche by gaining prana higher than the threshold prana required to surmount the barrier.

The definition of entropy yields the concept of time changing relentlessly as “Time’s arrow” with the following characteristics in probability theory

(1) Near a starting time of an experiment, the early data collected before the initial state is past, is macroscopically distinct from equilibrium. The system is overwhelmingly likely to evolve to greater disorder, toward equilibrium.

(2) In equilibrium, fluctuations have no sense of time so fluctuations remain within the standard deviation forever.

(3) Fluctuations from equilibrium to extremely unlikely states at the extreme of the probable range are extremely rare, virtually never happen or have vanishingly small probability.

Entropy Tends to Increase as the Duration of Time Increases due to Non-equilibrium

Isolated systems left alone tend to go to maximum entropy. Time and disorder tend to be correlated in change. All states of prana are equally possible at equilibrium. What is the mechanism for time and disorder to flow in the same direction? Most physical laws do not change whether time is going forward or back. It is only when large numbers of components are included that the sequence seems to irreversible.

Cause and Effect, Probability and Chaos

Chaos is characterized by unknown results unless extremely precise initial conditions and exact mathematical descriptions of cause and effect are known. Therefore,

1. In an isolated system, like a psyche during a short duration of time composed of vast numbers of UCC components, diffusions, which reduce entropy are exceedingly sensitive to initial conditions.

2. For a few components in “equilibrium” as defined above, the human does not observe that the overwhelming majority of diffusions reduce entropy. And any chaotic process will continue to be chaotic even if the initial conditions are altered slightly.

The notion that probability and chaos are elements of reality, as opposed to the belief that events can be determined exactly, suggests that most events are probable, even chaotic. The probability of events, rather than exact determination, led to the discovery of the third dimension of time, t6 (Howard& Kelleher, 2008a).


In addition to the probability measures used to describe parapsychological phenomena, such as the probable mean and the probable standard deviation, there is the probability of losing of memory of precognitive dreams with psychic content. The probability laws of quantum theory can be used to describe some neural processes.

It may be possible to use the definition of entropy in the psyche and probability mathematics to discover the origin of mental disorders. For example, use the description of entropy associated with disorder in the psyche, or use the non-equilibrium probability of entropy and disorder in the psyche or use he concepts of maximizing entropy as a result of minimizing order. Questions were asked which will be answered later.

The heuristics proposed herein can be applied to parapsychological research, as follows.

The conservation law for psychic content. Three hypotheses about combined physical and psychical concepts based on probability: the prana hypothesis, the psyche equilibrium as a time average of states hypothesis, and the diffusion of unconscious contents hypothesis.


The recommendation is to use the tools in the mathematics of probabilities which, when applied to the vast available data, will lead to general scientific attention to parapsychology.

Research is suggested by many questions raised herein. What are the exact math expressions shown as functions herein? Experiment with forms that could be substituted for these functional and verbal descriptions.

How does order in the psyche affect perception of extrasensory data? Under what condition does one perceive extra sensory data originating outside the psyche? What is the probability of determining whether one is perceiving information from outside the psyche without the normal five senses? What is a method of determining that one is perceiving without the normal five senses?

For the purpose becoming conscious of data not originated in the senses, or the psyche, is it better to have an ordered psyche or a completely disordered psyche? Explore the psychic laws and physical laws of diffusion of the UCC and conscious contents.

Psychiatrists would have another way of asking these questions, “Does extrasensory data diffuse into the conscious from a psyche in a disordered condition?” How to find a number to represent the randomness, the worst type of disorder, in a distribution of UCC components? What is the non-equilibrium probability of entropy and disorder in the psyche? What is the description of psyche components at a little less than equilibrium, what is the least ordered, and what components are completely ordered?

Could chaos theory be applied to human brains and minds to explain why they fall into harmony with one another? Does chaos theory explain how neurons in a brain become tuned to the psychic field, which vibrates at a given frequency?

The discovery of the third dimension of time, t6 should be related to parapsychological phenomena.

Appendix: Terminology of Probabilities

Observation = human senses an uncontrolled experiment

Trial = repetition of an experiment or observation under reasonably identical conditions.

Event = actual realized outcome of observation

Event -space = set of all possible distinct outcomes.

Mutually exclusive outcomes = division of event space into sectors such that each possible outcome belongs to only one sector.


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