# A Priori Probability – Definition

### A Priori Probability Definition

A priori Probability refers to the logical estimation of an incident’s probability. It can also be explained as a situation where one estimates a circumstance or current information about the position of something. This probability deals with independent event whereby the possibility of a given event happening is not in any way influenced by past events.

### A Little More on What is A Priori Probability

Priori probability calculation is often done through deductive reasoning. This is so because, in order to determine the number of the possible outcome of any occurence, one must apply logic. Also, this type of probability is based on prior information when making a conclusion.

### Examples of A Priori Probability

When tossing a coin, the possibility of it landing on a head or a tail in the second or subsequent toss is not in any way dependent on the first or previous result. Therefore, the chances of landing either side are equal hence the probability will definitely be 50%.

The probability can also apply to a firm’s earnings where whether the firm will make a profit, a loss or break even at any given year, the probability of each of the events happening is equal. In this case, the probability will, therefore, be ⅓ of each event occurring.

It is important to note that in Priori probability, experts draw a conclusion about the outcome of an event before looking at any statistics. Meaning the Priori probability is used to estimate the occurrence of an event before the event actually happens.

### Priori Probability Types

There are two types of Priori probability. The difference between the two probability is explained using Bayesian inference. They are as follows:

• Priori-This refers to capturing of general knowledge about a given statistics before looking at it. This means conclusions about the data is made before it is analyzed.
• Posteriori– This represents knowledge that includes the results. This means, in this type of probability, the outcome of an event is included in the conclusive results. Meaning the data about an event is analyzed before a conclusion is drawn.

### Priori Probability Uses

• The Priori probability estimates are applied on a nonlinear equation root during an evaluation.

• The main advantage of using this method of calculation is that the procedure has no assumptions.

### Limitation of Using Priori Probability in Calculation

• The major limitation about prior probability is that it only applies to a specific set of events (it is finite). For this reason, it can only be calculated for those events that are naturally independent. This is because the probability of most events happening is through conditioning to a certain percentage. The experiment is, therefore, not applicable where the outcomes are likely not to be equal.
• The experiment may fail especially when the figure of the possible experiment results is infinite (immeasurable).
• Difficult to estimate probabilities to the desired accuracy. You will need large sample sizes in order to get accurate results.

### Key Takeaways

• The conclusion is made prior knowledge or before analyzing any data.
• It involves deductive reasoning (logic deduction).
• It has a basic assumption that the results of a random experiment are likely to be equal.

### Academic Research on “ A Priori Probability”

The effect of a priori probability and complexity on decision making in a supervisory control task, Kerstholt, J. H., Passenier, P. O., Houttuin, K., & Schuffel, H. (1996). Human Factors, 38(1), 65-78. This study investigates how priori probability and complexity affects decision making in a supervisory control task i.e. their impact on the fault management and monitoring in a ship control task. The participants of the study supervised 4 independent shipping subsystems then were made to adjust them whenever they witnessed a deviation. They were also expected to diagnose the cause of the deviation in order to correct action; this was to be done through the consideration of additional subsystem information. The two variables, priori probability and complexity, were manipulated through varation of different disturbances that occur simultaneously. The findings of the study revealed that during the disturbance diagnosis, participants ignored the monitoring function. The authors also found cases of “cognitive lockup.” Furthermore, all the participants were found not to interrupt with any ongoing fault-finding process. Still, huge differences were noticed in the strategy selected and reasoning abilities.

Short-term recognition memory for pitch: Effect of a priori probability on response times and error rates, Coltheart, M., & Curthoys, I. (1968). Perception & Psychophysics, 4(2), 85-89. This study assesses the effects of a priority probability on error rates and response times using short-term recognition memory for pitch. According to the authors, when comparing 2 different pitch tones, the response of Ss is always faster when they respond “differently” than when they respond “same.” The authors found that the differential effects could be enhanced through the use of same trials more frequently. The responses, according to the experiment conducted, were also affected by the treatment. Thus, the authors discuss the results of latency and error using the RT model based upon a sequential-sampling. When the model was applied, the authors considered possible biasness towards the response “different.” The witnessed biasness was associated with the interstimulus intervals used in the experiment.

A simple mathematical proof of Boltzmann’s equal a priori probability hypothesis, Evans, D. J., Searles, D. J., & Williams, S. R. (2009). This paper provides an overview of the mathematical proof of Boltzmann’s equal a priori probability hypothesis. The authors use the Second Law Inequality and the Dissipation Theorem to offer a first-principle derivation of the postulate of equal a priori probability by Boltzmann. The authors illustrate how when the initial distribution differs from the uniform distribution over the energy hypersurface, the initial distribution relates to the uniform distribution. The findings are found to be analogous to the Botlzmann H-theorem. However, the result applies to dense fluids and dilute gases as well as permits a nonmonotonic relaxation to equilibrium. The authors also prove that uniform distribution is the only stationary, dissipationless distribution in the case of ergodic systems.

Monte Carlo analysis of resistive networks without a priori probability distributions, Barmish, B. R., & Kettani, H. (2000). In 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No. 00CH36353) (Vol. 3, pp. 263-266). IEEE. This paper analyses the resistive networks that do not have priori probability distributions. The authors have formulated and solved a new Monte Carlo problem based on a resistive network. They consider the way of finding the value of the resistors of the network without probability distribution. Using the assumption that there is no priory probability distributions for uncertain resistors, the authors were able to sought a certain type of ‘distribution robustness.’ Based on the findings, the authors particularize a new paradigm to circuits. Some of the performances obtained from the study differ considerably from the results of a more conventional simulation of Monte Carlo.

Lower limits of frequencies in computable sequences and relativized a priori probability, Muchnik, A. A. (1987). Theory of Probability and its Applications, 32(3), 513. This study assesses the lower limits of frequencies using relativized a priori probability and computable sequences.

A method of solving non-linear equations, using a priori probability estimates of the roots, Vysotskaya, I. N., & Strongin, R. G. E. (1983). USSR Computational Mathematics and Mathematical Physics, 23(1), 1-7. This article analyses and solves non-linear equations using a priori probability estimates. The authors use a priori probability estimates to describe algorithms for evaluating non-linear equation roots. Through indication of the maximum likelihood estimates of maximum likelihood, the authors distinguish the subintervals of the domain of specifications of the left-hand side of the equation. The formulation of the algorithms is based on the previously described approach. The authors obtain sufficient conditions for the algorithms to converge.