What is a Priori Probability?

Probability is a concept that is a universal asset. Whether it’s for finances or the simple roll of a dice, the probability is an active presence in most ‘games’ of chance. This article will explain describe the ‘priori probability and its various components.

What is it?

A conventional priori probability is an outcome that one calculates by way of logically examining a circumstance or pre-existing information in relation to a situation. It typically deals with independent events in which the likelihood of a given event occurring is not in any way different due to previous events. The most basic example of this concept would be an ordinary coin toss.

A concept that is close in relation to priori probability is ‘uninformative prior.’ Also known as a ‘diffuse prior’, these express vague or general information concerning a variable. They essentially signify “objective” information, like “The variable is positive” or “The variable is less than a certain level.”

By far the largest drawback with using priori probabilities as a way to define odds is that it is only useful for a finite set of events. This is due to the fact that most events are subject to ‘conditional probability’ to, at the very least, a small degree.

To provide a comparison, conditional probability is the likelihood of an event or outcome taking place. This is based on the occurrence of a past event or outcome. This brand of probability made clear by multiplying the probability of the preceding event by the updated probability of the consecutive – or the conditional – event.

Let’s say for example you have two events. The first event is that it is raining, and it has a 30% chance of rain today. The second event is that you will need to go outside, and that has a general probability of about 50%. A standard conditional probability would look at these events in relation to each other. This includes the probability that it’s both raining and you needing to go outside.

Breaking it down

The bulk of priori probabilities are useful within the deduction method of calculating. Traditionally, deduction means the subtraction of any item or expenditure from the overall gross income. This will effectively reduce the amount of income subject to income tax.

Tying this back to priori probabilities, this is due to the fact that you must use logic to properly determine the potential outcomes of an event. Doing so will determine the number of ways in which these outcomes can take place.

Much like what has been previously said, an example of this probability is flipping a coin. A regular, fair coin possesses two different sides. Each time you flip a coin, it has an equal chance of landing on either side, regardless of whatever the precious toss’ outcome was. The priori probability of landing on the “heads” side of the coin is, unsurprisingly, 50%.

An additional example is how the price of a share can change in any specific way. Its price can either increase, decrease, or even remain exactly the same. Ergo, according to a priori probability, it is safe to assume that there is a 1 in 3 (or 33%) chance of one of the outcomes occurring. Moreover, all else remains equal.

What about ‘prior’?

If you’re someone who has some basic knowledge about the different kinds of probabilities, you’re probably aware of the similar concept of ‘prior probability.’ There are concepts closely related to priori probabilities. This includes uninformative priors, which are more objective. However, prior probabilities are much broader in comparison. Contrary to initial assumptions, this broad probability is very much different from priori probabilities.

In regards to Bayesian statistical inference (a method in which Bayes’ theorem aids in updating the probability for a hypothesis), a prior probability is the likelihood of an event before the collection of new data. Many view this as being the best rational assessment of the probability of an outcome based on the current knowledge before performing an experiment.

The prior probability of an event will change every time new data or information becomes accessible. This way, it will produce a more accurate measure of a potential outcome. That rewritten probability becomes the ‘posterior probability’, which is then calculated using Bayes’ theorem. In statistical terms, the posterior probability is the likelihood of the first event taking place given the fact that the second event is already taking place.

Bayes’ theorem

Bayes’ theorem is a mathematical formula. Mathematician Thomas Bayes created this theorem, which is primarily for determining conditional probability. It also provides a way to be able to revise existing predictions or theories when given additional evidence. In terms of finances, Bayes’ theorem evaluates the risk of lending money to any potential borrowers.

It is not just in the financial field that this theorem comes in handy; in fact, it’s widespread. For example, it helps determine the accuracy of medical test results. This considers the likelihood of any given person having a disease and the general accuracy of the test. The theorem relies heavily on the incorporation of prior probability distributions in order to fully generate posterior probabilities.

Theorem formula

Bayes' Theorem

To further illustrate it:

P(A) is the probability of A occurring;

P(B) is the probability of B occurring;

P(A∣B) is the probability of A given B;

P(B∣A) is the probability of B given A; and

P(A⋂B)) is the probability of both A and B occurring.”

What it does

This formula is also a way to see how hypothetical new information affects the probability of an event occurring. This is supposing the new information turns out to be true. Investopedia editor, Adam Heyes, provides two examples below; one statistical and one numerical.

For instance, say a single card is drawn from a complete deck of 52 cards. The probability that the card is a king is 4 divided by 52, which equals 1/13 or approximately 7.69%. Remember that there are 4 kings in the deck. Now, suppose it is revealed that the selected card is a face card. The probability the selected card is a king, given it is a face card, is 4 divided by 12, or approximately 33.3%, as there are 12 face cards in a deck.”

As a numerical example, imagine there is a drug test that is 98% accurate, meaning 98% of the time it shows a true positive result for someone using the drug and 98% of the time it shows a true negative result for nonusers of the drug. Next, assume 0.5% of people use the drug. If a person selected at random tests positive for the drug, the following calculation can be made to see whether the probability the person is actually a user of the drug.

(0.98 x 0.005) / [(0.98 x 0.005) + ((1 – 0.98) x (1 – 0.005))] = 0.0049 / (0.0049 + 0.0199) = 19.76%

Regarding the drug scenario, Bayes’ theorem shows that even if a person were to test positive, it is much more likely that the person does not use the drug.


There are many layers to priori probability that extend to other concepts. To better understand it means needing to understand the elements that surround it and are related to it.

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