What it means
The phrase a priori means from what comes before, that is, from reasoning rather than from experience. In probability it describes an estimate deduced from a known structure, such as the six equally likely faces of a fair die or the 52 cards in a shuffled deck.
The method only works when the outcomes really are equally likely and can be listed completely. Cards, dice, coins and lottery draws qualify, while customer behaviour, loan defaults and machine breakdowns do not, because their outcomes are neither equally likely nor fully known in advance.
In finance the honest use of a priori probability is narrow but real. It underpins the pricing of purely random draws such as prize bonds and raffles, and it provides the baseline against which observed results are judged, because a process that should produce an outcome 25% of the time yet produces it 60% of the time over many trials is not behaving as assumed.
Analysts distinguish three routes to a probability. A priori reasoning uses structure, empirical or frequentist probability uses observed frequencies from history, and subjective probability uses informed judgement, which is what most business forecasts actually rest on.
The danger is dressing up a subjective guess as an a priori certainty. Saying a new product has a 50% chance of success because it either works or it does not misuses the method, since those two outcomes are plainly not equally likely.
In practice
Real-world examples.
Example
An internal audit team selects 4 invoices at random from a batch of 200 for detailed testing. Before any testing begins, the a priori probability that any specific invoice is chosen is 4 / 200 = 2%, which the team documents to show the sample was genuinely random.
Example
A fund manager explains to a client that the chance of a fair coin landing heads five times in a row is 1 / 32, or about 3.1%. He uses it to show why a run of five good quarters is not automatically evidence of skill.
Example
An insurer designs a promotional scratch card with 500,000 cards and 250 prizes of $200 each. The a priori win probability is 250 / 500,000 = 0.05%, and the total prize cost is 250 x $200 = $50,000, which marketing can budget with certainty because the odds are fixed by design.
Formula
Calculation
A priori probability = Number of favourable outcomes / Total number of equally likely outcomes
A company runs a staff prize draw. It sells 20,000 tickets at $5 each, and one ticket wins a single prize of $50,000.
The probability that any given ticket wins is 1 / 20,000 = 0.00005, or 0.005%.
The expected value of a ticket is the prize multiplied by the probability of winning, which is $50,000 / 20,000 = $2.50.
Because a ticket costs $5 and is worth $2.50 in expectation, each ticket carries an expected loss to the buyer of $5.00 - $2.50 = $2.50. The draw takes in 20,000 x $5 = $100,000 and pays out $50,000, leaving $50,000 for the charity it supports.
A simpler illustration: a company has 12 equally sized sales territories, of which 3 are coastal. If an auditor picks one territory at random, the a priori probability of picking a coastal one is 3 / 12 = 0.25, or 25%.Case study
Seen in the real world.
Ardent Mutual is an invented, illustrative savings provider used here to show a priori probability applied well and then misapplied. It launched a prize savings account in which each $100 held earned one entry into a monthly draw, with 400 prizes of $500 shared across 2,000,000 entries.
The a priori arithmetic was straightforward. Each entry had a 400 / 2,000,000 = 0.02% chance of winning, and the total monthly prize fund was 400 x $500 = $200,000, giving an expected return per entry of $200,000 / 2,000,000 = $0.10 a month. That is $1.20 a year on each $100 held, an effective annual rate of 1.2%, and the product team could quote it confidently because it followed from the design of the draw rather than from any forecast.
The mistake in this fictional case came later, when the same team used a priori reasoning to claim a 50% chance that the product would hit its customer target, on the grounds that it either would or would not. Those are not equally likely outcomes, and when take-up reached only 30% of target the board began insisting that behavioural forecasts be based on observed data instead.
Watch out
Common mistakes.
- Applying a priori reasoning to events whose outcomes are not equally likely, such as treating a product launch as if it were a coin flip.
- Confusing the probability of a single event with the pattern of a long run, and expecting a fair coin to correct itself after several heads.
- Forgetting that the method assumes the mechanism is fair, so it says nothing useful about a loaded die or a rigged draw.
Questions
People also ask.
How is a priori probability different from empirical probability?
A priori is deduced from the structure of the situation before any data exists, while empirical probability is measured from what has actually happened.
Is a priori probability useful in business forecasting?
Only in narrow cases such as random draws and sampling design, since most forecasting relies on historical frequencies or informed judgement instead.
What is the link to expected value?
Expected value multiplies each outcome by its probability, so an a priori probability supplies the weighting whenever the outcomes are equally likely by design.
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